An introduction to some aspects of functional analysis

Stephen Semmes Rice University

Abstract These informal notes deal with some very basic objects in functional analysis, including norms and seminorms on vector spaces, bounded linear operators, and dual spaces of bounded linear functionals in particular.

Contents

1 Norms on vector spaces 3 2 Convexity 4 3 Inner product spaces 5 4 A few simple estimates 7 5 Summable functions 8 6 p-Summable functions 9 7 p = 2 10 8 Bounded continuous functions 11 9 Integral norms 12 10 Completeness 13 11 Bounded linear mappings 14 12 Continuous extensions 15 13 Bounded linear mappings, 2 15 14 Bounded linear functionals 16 15 ℓ^{1}(E)^{∗} 18 16 c_{0}(E)^{∗} 18 17 Hölder’s inequality 20 18 ℓ^{p}(E)^{∗} 20 19 Separability 22 20 An extension lemma 22 21 The Hahn–Banach theorem 23 22 Complex vector spaces 24 23 The weak^{∗} topology 24 24 Seminorms 25 25 The weak^{∗} topology, 2 26 26 The Banach–Alaoglu theorem 27 27 The weak topology 28 28 Seminorms, 2 28 29 Convergent sequences 30 30 Uniform boundedness 31 31 Examples 32 32 An embedding 33 33 Induced mappings 33 34 Continuity of norms 34 35 Infinite series 35 36 Some examples 36 37 Quotient spaces 37 38 Quotients and duality 38 39 Dual mappings 39 40 Second duals 39

41 Continuous functions 41

42 Bounded sets 42

43 Hahn–Banach, revisited 43

References 44

1 Norms on vector spaces

Let V be a vector space over the real numbers R or the complex numbers C. A norm on V is a nonnegative real-valued function ∥v∥ defined for v ∈ V such that ∥v∥ = 0 (1.1)

if and only if v = 0, ∥t v∥ = |t| ∥v∥ (1.2)

for every v ∈ V and t ∈ R or C, as appropriate, and

∥v + w∥ ≤ ∥v∥ + ∥w∥ (1.3)

for every v, w ∈ V. Here |t| is the absolute value of t when t ∈ R, and the modulus of t when t ∈ C. For example, if V is the one-dimensional vector space R or C, then the absolute value or modulus defines a norm on V, and every norm on V is a positive multiple of this norm.

More generally, let n be a positive integer, and let V be the space R^{n} or C^{n} of n-tuples v = (v_{1}, …, v_{n}) of real or complex numbers. As usual, this is a vector space with respect to coordinatewise addition and scalar multiplication. The standard Euclidean norm on V is given by

∥v∥ = ( ∑_{j=1}^{n} |v_{j}|^{2})^{1/2}. (1.4)

It is well-known that this is a norm, although the triangle inequality (1.3) is not immediately obvious. A couple of ways to show the triangle inequality in this case will be reviewed in the next sections.

It is easy to see directly that

∥v∥_{1} = ∑_{j=1}^{n} |v_{j}| (1.5)

satisfies the triangle inequality. Hence ∥v∥_{1} defines a norm on R^{n}, C^{n}. Similarly,

∥v∥_{∞} = max(|v_{1}|, …, |v_{n}|) (1.6) defines a norm on R^{n}, C^{n}. The standard Euclidean norm on R^{n}, C^{n} is sometimes denoted ∥v∥_{2}.

If ∥v∥ is a norm on a real or complex vector space V, then

d(v, w) = ∥v − w∥ (1.7)

defines a metric on V. For instance, the standard Euclidean metric on R^{n} or C^{n} is the same as the metric associated to the standard Euclidean norm in this way. The metrics on R^{n} or C^{n} associated to the norms ∥v∥_{1}, ∥v∥_{∞} determine the same topology as the standard Euclidean metric, and we shall say more about this soon. The main point is that ∥v∥_{1}, ∥v∥_{2}, and ∥v∥_{∞} are equivalent norms on R^{n} and C^{n} for each positive integer n, in the sense that each is bounded by constant multiples of the others.

2 Convexity

Let V be a real or complex vector space. As usual, a set E ⊆ V is said to be convex if for every x, y ∈ E and real number t with 0 < t < 1, we have that

t x + (1 − t) y (2.1)

is also an element of E. If ∥v∥ is a norm on V, then the closed unit ball

B_{1} = {v ∈ V : ∥v∥ ≤ 1} (2.2)

is a convex set in V. Similarly, the open unit ball

{v ∈ V : ∥v∥ < 1} (2.3)

is also a convex set in V.

Conversely, suppose that ∥v∥ is a nonnegative real-valued function on V which is positive when v ≠ 0 and satisfies the homogeneity condition (1.2). If the corresponding closed unit ball B_{1} is convex, then one can check that ∥v∥ satisfies the triangle inequality, and is thus a norm. For if v, w are nonzero vectors in V, then

v′ = \frac{v}{∥v∥}, w′ = \frac{w}{∥w∥} (2.4)

have norm 1, and the convexity of B_{1} implies that

\frac{∥v∥}{∥v∥ + ∥w∥} v′ + \frac{∥w∥}{∥v∥ + ∥w∥} w′ (2.5)

has norm less than or equal to 1. Equivalently,

\frac{v + w}{∥v∥ + ∥w∥} ∈ B_{1}, (2.6)

which implies (1.3).

Let n be a positive integer, let p be a positive real number, and consider

∥v∥_{p} = ( ∑_{j=1}^{n} |v_{j}|^{p})^{1/p} (2.7)

for v ∈ R^{n} or C^{n}. This is the same as the standard Euclidean norm when p = 2, and is easily seen to be a norm when p = 1, as in the previous section. Let us check that the closed unit ball associated to ∥v∥_{p} is convex when p > 1, so that ∥v∥_{p} defines a norm in this case. This does not work for p < 1, even when n = 2.

Suppose that v, w ∈ R^{n} or C^{n} satisfy ∥v∥_{p}, ∥w∥_{p} ≤ 1, which is the same as

∑_{j=1}^{n} |v_{j}|^{p}, ∑_{j=1}^{n} |w_{j}|^{p} ≤ 1. (2.8)

If p ≥ 1 and 0 < t < 1, then

|t v_{j} + (1 − t) w_{j}|^{p} ≤ (t |v_{j}| + (1 − t) |w_{j}|)^{p} ≤ t |v_{j}|^{p} + (1 − t) |w_{j}|^{p} (2.9)

for each j, because of the convexity of the function f(x) = x^{p} on the positive real numbers. Summing over j, we get that

∑_{j=1}^{n} |t v_{j} + (1 − t) w_{j}|^{p} ≤ t ∑_{j=1}^{n} |v_{j}|^{p} + (1 − t) ∑_{j=1}^{n} |w_{j}|^{p} ≤ 1. (2.10)

This implies that ∥t v + (1 − t) w∥_{p} ≤ 1, (2.11)

as desired.

3 Inner product spaces

An inner product on a real or complex vector space V is a real or complex-valued function ⟨v, w⟩ defined for v, w ∈ V, as appropriate, which satisfies the following conditions. First, ⟨v, w⟩ is linear as a function of v for each w ∈ V. Second,

⟨w, v⟩ = ⟨v, w⟩ (3.1)

for every v, w ∈ V in the real case, while

⟨w, v⟩ = \overline{⟨v, w⟩} (3.2)

in the complex case. Here \overline{z} denotes the complex conjugate of z ∈ C, defined by \overline{z} = x − y i (3.3) when z = x + y i and x, y ∈ R. It follows from this symmetry condition that ⟨v, w⟩ is linear in w in the real case, and conjugate-linear in w in the complex case. It also follows that ⟨v, v⟩ ∈ R (3.4)

for every v ∈ V, even in the complex case. The third requirement of an inner product is that ⟨v, v⟩ > 0 (3.5)

for every v ∈ V with v ≠ 0, and of course the inner product is equal to 0 when v = 0.

If ⟨v, w⟩ is an inner product on V, then the Cauchy–Schwarz inequality states that |⟨v, w⟩| ≤ ⟨v, v⟩^{1/2} ⟨w, w⟩^{1/2} (3.6)

for every v, w ∈ V. Because of homogeneity, it suffices to show that

|⟨v, w⟩| ≤ 1 (3.7)

when ⟨v, v⟩ = ⟨w, w⟩ = 1, (3.8)

since the inequality is trivial when v or w is 0. To do this, one can use the fact that ⟨v + t w, v + t w⟩ ≥ 0 (3.9)

for every t ∈ R or C, and expand the inner product to estimate ⟨v, w⟩. The norm on V associated to the inner product is defined by

∥v∥ = ⟨v, v⟩^{1/2}. (3.10)

This clearly satisfies the positivity and homogeneity requirements of a norm. In order to verify the triangle inequality, one can use the Cauchy–Schwarz inequality to get that

(3.11)  ∥v + w∥^{2} = ⟨v + w, v + w⟩
= ⟨v, v⟩ + ⟨v, w⟩ + ⟨w, v⟩ + ⟨w, w⟩
≤ ∥v∥^{2} + 2 ∥v∥ ∥w∥ + ∥w∥^{2} = (∥v∥ + ∥w∥)^{2}

for every v, w ∈ V.

The standard inner product on R^{n} is defined by

⟨v, w⟩ = ∑_{j=1}^{n} v_{j} w_{j}. (3.12)

Similarly, the standard inner product on C^{n} is defined by

⟨v, w⟩ = ∑_{j=1}^{n} v_{j} \overline{w_{j}}. (3.13)

The corresponding norms are the standard Euclidean norms.

4 A few simple estimates

Observe that ∥v∥_{∞} ≤ ∥v∥_{p} (4.1) for each v ∈ R^{n} or C^{n} and p > 0. If 0 < p < q < ∞, then

∑_{j=1}^{n} |v_{j}|^{q} ≤ ∥v∥_{∞}^{q−p} ∑_{j=1}^{n} |v_{j}|^{p} (4.2)

for every v ∈ R^{n} or C^{n}, and hence ∥v∥_{q} ≤ ∥v∥_{∞}^{1−(p/q)} ∥v∥_{p}^{p/q} ≤ ∥v∥_{p}. (4.3) Thus ∥v∥_{p} is monotone decreasing in p.

In the other direction, ∥v∥_{p} ≤ n^{1/p} ∥v∥_{∞} (4.4) for each v ∈ R^{n} or C^{n} and p > 0. If 0 < p < q < ∞, then ∥v∥_{p} ≤ n^{(1/p)−(1/q)} ∥v∥_{q}. (4.5) To see this, remember that f(x) = x^{r} is a convex function of x ≥ 0 when r ≥ 1, so that

(\frac{1}{n} ∑_{j=1}^{n} a_{j})^{r} ≤ \frac{1}{n} ∑_{j=1}^{n} a_{j}^{r} (4.6)

for any nonnegative real numbers a_{1}, …, a_{n}. Applying this with r = q/p, we get that

(\frac{1}{n} ∑_{j=1}^{n} |v_{j}|^{p})^{q/p} ≤ \frac{1}{n} ∑_{j=1}^{n} |v_{j}|^{q}, (4.7)

or

(\frac{1}{n} ∑_{j=1}^{n} |v_{j}|^{p})^{1/p} ≤ (\frac{1}{n} ∑_{j=1}^{n} |v_{j}|^{q})^{1/q}, (4.8)

as desired.

The fact that ∥v∥_{1} ≤ ∥v∥_{p} (4.9) when 0 < p < 1 and n = 2 can be re-expressed as (a + b)^{p} ≤ a^{p} + b^{p} (4.10) for every a, b ≥ 0. It follows that

∑_{j=1}^{n} |v_{j} + w_{j}|^{p} ≤ ∑_{j=1}^{n} |v_{j}|^{p} + ∑_{j=1}^{n} |w_{j}|^{p} (4.11)

for every v, w ∈ R^{n} or C^{n} when 0 < p < 1, and hence ∥v + w∥_{p}^{p} ≤ ∥v∥_{p}^{p} + ∥w∥_{p}^{p}. (4.12) This is a natural substitute for the triangle inequality in this case.

5 Summable functions

Let E be a nonempty set, and let f(x) be a nonnegative real-valued function on E. If E has only finitely many elements, then the sum

∑_{x∈E} f(x) (5.1)

is defined in the usual way. Otherwise, (5.1) is defined to be the supremum of

∑_{x∈E_{1}} f(x) (5.2)

over all finite subsets E_{1} of E, which may be +∞. If there is a finite upper bound for the subsums (5.2), so that (5.1) is finite, then we say that f is summable on E. Note that f is summable on E if there is a nonnegative summable function F on E such that f(x) ≤ F (x) for every x ∈ E.

If f_{1}, f_{2} are nonnegative functions on E, then

∑_{x∈E}(f_{1}(x) + f_{2}(x)) = ∑_{x∈E} f_{1}(x) + ∑_{x∈E} f_{2}(x). (5.3)

This is easy to see, with the standard convention that

a + ∞ = ∞ + a = ∞, 0 ≤ a ≤ ∞. (5.4)

In particular, f_{1} + f_{2} is summable on E when f_{1} and f_{2} are summable on E. Similarly, if f is a nonnegative function on E and t ≥ 0, then

∑_{x∈E} t f(x) = t ∑_{x∈E} f(x). (5.5)

This uses the conventions that t · ∞ = ∞ when t > 0, and 0 · ∞ = 0.

A real or complex-valued function f on E is said to be summable on E if |f(x)| is summable on E. It follows from the previous remarks that the summable functions on E form a linear subspace of the vector space of real or complex-valued functions on E, with respect to pointwise addition and scalar multiplication. If f is a summable function on E and ϵ > 0, then the set

E_{ϵ}(f) = {x ∈ E : |f(x)| ≥ ϵ} (5.6)

has only finitely many elements, and hence

E(f) = {x ∈ E : f(x) ≠ 0} (5.7)

has only finitely or countably many elements. More precisely,

|E_{ϵ}(f)| ≤ \frac{1}{ϵ} ∑_{x∈E} |f(x)|, (5.8)

where |E_{ϵ}(f)| denotes the number of elements of E_{ϵ}(f).

If f is a real or complex-valued summable function on E, then the sum (5.1) can be defined in various equivalent ways. One way is to express f as a linear combination of nonnegative summable functions, and then use the definition of the sum in that case. Another way is to enumerate the elements of E(f) in a sequence, and treat the sum as an infinite series. A basic property of the sum is that

|∑_{x∈E} f(x)| ≤ ∑_{x∈E} |f(x)| (5.9)

for any summable function f on E. As usual, the sum of a summable function is also linear in the function.

6 p-Summable functions

Let E be a nonempty set, and let p be a positive real number. A real or complexvalued function f on E is said to be p-summable if |f(x)|^{p} is a summable function on E. Thus p-summability is the same as summability when p = 1. One can check that the spaces ℓ^{p}(E, R), ℓ^{p}(E, C) (6.1)

of p-summable real and complex-valued functions on E are linear subspaces of the vector spaces of all such functions on E. This can be extended to p = ∞ by taking ℓ^{∞}(E, R), ℓ^{∞}(E, C) to be the spaces of bounded real and complex-valued functions on E.

If f is a p-summable function on E for some p, 0 < p < ∞, then we put

∥f∥_{p} = ( ∑_{x∈E} |f(x)|^{p})^{1/p}. (6.2)

Here the sum over E is defined to be the supremum over all finite subsums, as in the previous section. Similarly, if f is a bounded function on E, then we put

∥f∥_{∞} = sup{|f(x)| : x ∈ E}. (6.3)

It is easy to see that ∥f_{1} + f_{2}∥_{p} ≤ ∥f_{1}∥_{p} + ∥f_{2}∥_{p} (6.4)

when f_{1}, f_{2} ∈ ℓ^{p}(E, R) or ℓ^{p}(E, C) and 1 ≤ p ≤ ∞, so that ∥f∥_{p} defines a norm on these spaces in this case. This is quite straightforward when p = 1, ∞, and can be obtained from the corresponding statement for finite sums when 1 < p < ∞. If 0 < p < 1 and f_{1}, f_{2} are p-summable functions on E, then

∥f_{1} + f_{2}∥_{p}^{p} ≤ ∥f_{1}∥_{p}^{p} + ∥f_{2}∥_{p}^{p}. (6.5)

This can be derived using the same argument as for finite sums.

If f is p-summable, 0 < p < ∞, then f is bounded, and

∥f∥_{∞} ≤ ∥f∥_{p}. (6.6)

Moreover, f is q-summable for each q > p, and

∥f∥_{q} ≤ ∥f∥_{p}. (6.7)

This can be shown in the same way as for finite sums.

A function f on E is said to “vanish at infinity” if for each ϵ > 0 there are only finitely many x ∈ E such that |f(x)| ≥ ϵ. The spaces

c_{0}(E, R), c_{0}(E, C) (6.8)

of real and complex-valued functions vanishing at infinity on E are linear subspaces of the corresponding spaces of bounded functions on E. These are proper linear subspaces when E has infinitely many elements, and otherwise every function on E is bounded and vanishes at infinity when E has only finitely many elements. As in the previous section, p-summable functions vanish at infinity for each 0 < p < ∞.

7 p = 2

It is well known that

a b ≤ \frac{a^{2} + b^{2}}{2} (7.1)

for every a, b ≥ 0, since

0 ≤ (a − b)^{2} = a^{2} − 2 a b + b^{2}. (7.2)

If f_{1}, f_{2} are square-summable functions on a nonempty set E, then it follows that

|f_{1}(x)| |f_{2}(x)| ≤ \frac{|f_{1}(x)|^{2}}{2} + \frac{|f_{2}(x)|^{2}}{2} (7.3)

for every x ∈ E, and hence

∑_{x∈E} |f_{1}(x)| |f_{2}(x)| ≤ \frac{1}{2} ∑_{x∈E} |f_{1}(x)|^{2} + \frac{1}{2} ∑_{x∈E} |f_{2}(x)|^{2} < ∞. (7.4)

This implies that the product f_{1}(x) f_{2}(x) is summable on E. Alternatively, one can get the same conclusion using the Cauchy–Schwarz inequality for finite sums to show that

∑_{x∈E} |f_{1}(x)| |f_{2}(x)| ≤ ( ∑_{x∈E} |f_{1}(x)|^{2})^{1/2} ( ∑_{x∈E} |f_{2}(x)|^{2})^{1/2}. (7.5)

At any rate, this permits us to define an inner product on ℓ^{2}(E, R) by

⟨f_{1}, f_{2}⟩ = ∑_{x∈E} f_{1}(x) f_{2}(x), (7.6)

and on ℓ^{2}(E, C) by

⟨f_{1}, f_{2}⟩ = ∑_{x∈E} f_{1}(x) \overline{f_{2}(x)}. (7.7)

These are basically the same as the standard inner products on R^{n} and C^{n} when E is a finite set with n elements. The norms associated to these inner products are the same as ∥f∥_{2} defined in the previous section.

8 Bounded continuous functions

Let X be a topological space, and let

C_{b}(X, R), C_{b}(X, C) (8.1)

be the spaces of bounded continuous real or complex-valued functions on X. As usual, these are vector spaces with respect to pointwise addition and scalar multiplication. If X is compact, then every continuous function on X is bounded, and we may drop the subscript b. The supremum norm of a bounded continuous function f is defined by

∥f∥_{∞} = sup{|f(x)| : x ∈ X}. (8.2)

It is easy to see that this defines a norm on these vector spaces. Of course, constant functions on X are automatically bounded and continuous. If X is equipped with the discrete topology, then every function on X is continuous.

Locally compact Hausdorff spaces form a nice class of topological spaces. If X is such a space, K ⊆ X is compact, and V ⊆ X is an open set with K ⊆ V, then it is well known that there is a continuous real-valued function f on X such that f(x) = 1 (8.3)

when x ∈ K, f(x) = 0 (8.4)

when x ∈ X\V, and 0 ≤ f(x) ≤ 1 (8.5)

for every x ∈ X. In particular, there are plenty of bounded continuous functions on X in this case.

A continuous function f on X is said to “vanish at infinity” if for every ϵ > 0 there is a compact set K ⊆ X such that

|f(x)| < ϵ (8.6)

for every x ∈ X\K. If X is equipped with the discrete topology, then the compact subsets of X are the same as the finite sets, and this definition reduces to the previous one in the context of summable functions. For any X, the space of real or complex-valued continuous functions on X that vanish at infinity may be denoted C_{0}(X, R), C_{0}(X, C). (8.7)

It is easy to see that these are linear subspaces of the corresponding spaces of bounded continuous functions on X, since continuous functions are bounded on compact sets. If X is compact, then every continuous function on X vanishes at infinity trivially.

The support of a continuous function f on X is the closure of the set where f ≠ 0, i.e., supp f = \overline{\{x ∈ X : f(x) ≠ 0\}}. (8.8)

The spaces of real or complex-valued continuous functions on X with compact support may be denoted

C_{00}(X, R), C_{00}(X, C). (8.9)

These are linear subspaces of the corresponding spaces of continuous functions on X that vanish at infinity, and are the same as the spaces of all continuous functions on X when X is compact. There are a lot of continuous functions with compact support on locally compact Hausdorff spaces, as before.

9 Integral norms

Let f be a continuous real or complex-valued function on the closed unit interval [0, 1] in the real line. For each positive real number p, put

∥f∥_{p} = ( ∫_{0}^{1} |f(x)|^{p} dx)^{1/p}. (9.1)

This obviously satisfies the triangle inequality when p = 1, and one can show that it also holds for p > 1 using the same argument as for finite sums. Thus ∥f∥_{p} defines a norm on the vector space of continuous functions on [0, 1] when p ≥ 1. If 0 < p < 1, then

∥f_{1} + f_{2}∥_{p}^{p} ≤ ∥f_{1}∥_{p}^{p} + ∥f_{2}∥_{p}^{p} (9.2)

for all continuous functions f_{1}, f_{2} on [0, 1]. We also have that

∥f∥_{p} ≤ ∥f∥_{∞} (9.3)

for each continuous function f on [0, 1] and every p > 0, where ∥f∥_{∞} is the supremum norm of f, as in the previous section. If 0 < p < q < ∞, then one can check that ∥f∥_{p} ≤ ∥f∥_{q}, (9.4)

using convexity as in Section 4.

Now let f be a continuous real or complex-valued function on the real line with compact support. For each p > 0, consider

∥f∥_{p} = ( ∫_{−∞}^{∞} |f(x)|^{p} dx)^{1/p}. (9.5)

More precisely, the integral here can be reduced to one on a bounded interval, since f has compact support. As before, this defines a norm on the vector space of continuous functions on R with compact support when p ≥ 1, and satisfies the usual substitute for the triangle inequality when 0 < p < 1. However, ∥f∥_{p} is normally neither monotone increasing nor decreasing in p in this case.

If p = 2, then these norms associated to suitable inner products. On the unit interval, these inner products are given by

⟨f_{1}, f_{2}⟩ = ∫_{0}^{1} f_{1}(x) f_{2}(x) dx (9.6)

in the real case, and

⟨f_{1}, f_{2}⟩ = ∫_{0}^{1} f_{1}(x) \overline{f_{2}(x)} dx (9.7)

in the complex case. The inner products on continuous functions with compact support on R are defined similarly.

Of course, there are analogous norms and inner products for other situations in which integration is defined. This includes sums over discrete sets as another special case.

10 Completeness

Remember that a metric space (M, d(x, y)) is said to be complete if every Cauchy sequence in M converges to an element of M. If M is complete and N ⊆ M, then N is complete with respect to the restriction of d(x, y) to x, y ∈ N if and only if N is a closed set in M.

Let V be a real or complex vector space equipped with a norm and hence a metric, as in Section 1. If V is complete with respect to this metric, then V is said to be a Banach space. If the norm is also associated to an inner product, then V is said to be a Hilbert space.

It is well known that the real line is complete with respect to the standard metric, and that consequently R^{n} and C^{n} are complete for each n. One can use this to show that the ℓ^{p}(E) spaces are complete with respect to the p-norm ∥f∥_{p} for every nonempty set E and 1 ≤ p ≤ ∞. For if {f_{j}}_{j=1}^{∞} is a Cauchy sequence of functions on E with respect to the p-norm for some p, then {f_{j}(x)}_{j=1}^{∞} is a Cauchy sequence of real or complex numbers for each x ∈ E. Hence {f_{j}}_{j=1}^{∞} converges pointwise to a function f on E, and one can use the fact that the norms ∥f_{j}∥_{p} are uniformly bounded in j to show that f ∈ ℓ^{p}(E). Using the Cauchy condition for {f_{j}}_{j=1}^{∞} in ℓ^{p}(E), one can also show that {f_{j}}_{j=1}^{∞} converges to f in ℓ^{p}(E). Similarly, ℓ^{p}(E) is complete when 0 < p < 1, with respect to the metric ∥f_{1} − f_{2}∥_{p}^{p}. (10.1)

In addition, one can check that c_{0}(E) is a closed subspace of ℓ^{∞}(E), and is thus complete with respect to the supremum norm.

The space C_{b}(X) of bounded continuous functions on any topological space X is complete with respect to the supremum norm. As in the previous paragraph, one can first show that a Cauchy sequence in C_{b}(X) converges pointwise, and then use the Cauchy condition with respect to the supremum norm to check that the sequence converges uniformly, and that the limit is also bounded and continuous. The space C_{0}(X) of continuous functions that vanish at infinity on X is a closed subspace of C_{b}(X), and hence also complete with respect to the supremum norm. However, the space of continuous functions on the unit interval is not complete with respect to the integral norm ∥f∥_{p} for any p, 1 ≤ p < ∞, nor with respect to the metric ∥f_{1} − f_{2}∥_{p}^{p} when 0 < p < 1. The completions of these spaces can be described in terms of the Lebesgue integral.

11 Bounded linear mappings

Let V and W be vector spaces, both real or both complex, and equipped with norms ∥ · ∥_{V}, ∥ · ∥_{W}, respectively. A linear mapping T : V → W is said to be bounded if there is an A ≥ 0 such that

∥T (v)∥_{W} ≤ A ∥v∥_{V} (11.1)

for every v ∈ V. It is easy to see that every linear mapping is bounded when V is R^{n} or C^{n} for some positive integer n. If V = W with the same norm, then the identity mapping I(v) = v is bounded, with A = 1.

If T is a linear mapping from V into W that satisfies (11.1), then

∥T (v_{1}) − T (v_{2})∥_{W} ≤ A ∥v_{1} − v_{2}∥_{V} (11.2)

for every v_{1}, v_{2} ∈ V, and it follows that T : V → W is uniformly continuous. Conversely, if a linear mapping T : V → W is continuous at the origin, then it is bounded. For if there is a δ > 0 such that

∥T (v)∥_{W} < 1 (11.3)

when v ∈ V satisfies ∥v∥_{V} < δ, then (11.1) holds with A = 1/δ.

A linear mapping from a real or complex vector space V into the real or complex numbers, as appropriate, is also known as a linear functional on V. A linear functional on a vector space V with a norm ∥ · ∥_{V} is bounded if it is bounded with respect to the standard norm on R or C.

Remember that the kernel of a linear mapping T : V → W is the linear subspace of V consisting of the v ∈ V such that T (v) = 0. By continuity, the kernel of a bounded linear mapping is a closed subspace of V. Conversely, one can show that a linear functional on V is bounded when its kernel is closed.

12 Continuous extensions

Let (M, d(x, y)) and (N, ρ(u, v)) be metric spaces. If N is complete, then every uniformly continuous mapping from a dense subset of M into N has a unique extension to a uniformly continuous mapping from M into N. Ordinary continuity is sufficient for uniqueness of the extension, but uniform continuity is needed for its existence.

Now let V and W be vector spaces, both real or both complex, and equipped with norms. If W is complete, then every bounded linear mapping from a dense linear subspace of V into W has a unique extension to a bounded linear mapping from V into W. Bounded linear mappings are automatically uniformly continuous, as in the previous section, and the remaining point to check is that the extension is also linear in this case. One can use the same constant A as in (11.1) for the extension as for the initial mapping too.

For example, let E be an infinite set, and let V be the space of real or complex-valued summable functions on E. It is easy to see that the space of functions f on E such that f(x) ≠ 0 for only finitely many x ∈ E is dense in ℓ^{1}(E) with respect to the ℓ^{1} norm. For each f in this subspace, the sum

∑_{x∈E} f(x) (12.1)

can be defined in the obvious way, and satisfies

|∑_{x∈E} f(x)| ≤ ∑_{x∈E} |f(x)| = ∥f∥_{1}. (12.2)

Hence there is a unique extension of (12.1) to a bounded linear functional on ℓ^{1}(E). This gives another approach to the definition of the sum of a summable function, as in Section 5.

13 Bounded linear mappings, 2

Let V and W be vector spaces, both real or both complex. The space L(V, W) of linear mappings from V into W is also a vector space in a natural way, since the sum of two linear mappings from V into W is also a linear mapping, as is the product of a linear mapping with a real or complex number, as appropriate. If V and W are equipped with norms ∥ · ∥_{V} and ∥ · ∥_{W}, respectively, then one can check that the space BL(V, W) of bounded linear mappings from V into W is a linear subspace of L(V, W).

The operator norm of a bounded linear mapping T : V → W is defined by

∥T ∥_{op} = sup{∥T (v)∥_{W} : v ∈ V, ∥v∥_{V} ≤ 1}. (13.1)

Equivalently, ∥T (v)∥_{W} ≤ ∥T ∥_{op} ∥v∥_{V} (13.2) for every v ∈ V, and ∥T ∥_{op} is the smallest number with this property, which is to say the smallest value of A for which (11.1) holds. It is easy to see that the operator norm defines a norm on the vector space BL(V, W). For example, if V = W with the same norm, then the identity mapping I(v) = v has operator norm equal to 1.

Suppose that V_{1}, V_{2}, and V_{3} are vector spaces, all real or all complex, and equipped with norms. If T_{1} : V_{1} → V_{2} and T_{2} : V_{2} → V_{3} are bounded linear mappings, then their composition T_{2} ∘ T_{1} : V_{1} → V_{3}, defined by

(T_{2} ∘ T_{1})(v) = T_{2}(T_{1}(v)), v ∈ V, (13.3)

is also a bounded linear mapping. Moreover,

∥T_{2} ∘ T_{1}∥_{op,13} ≤ ∥T_{1}∥_{op,12} ∥T_{2}∥_{op,23}, (13.4)

where the subscripts indicate which spaces are involved in the operator norms.

If W is complete, then one can show that the space of bounded linear mappings from V into W is complete with respect to the operator norm. For if {T_{j}}_{j=1}^{∞} is a Cauchy sequence in BL(V, W), then {T_{j}(v)}_{j=1}^{∞} is a Cauchy sequence in W for each v ∈ V, and hence converges in W. The limit determines a linear mapping T : V → W, and one can use the boundedness of the operator norms of the T_{j}’s to show that T is bounded too. Using the Cauchy condition again, one can show that {T_{j}}_{j=1}^{∞} converges to T in the operator norm, as desired.

14 Bounded linear functionals

Let V be a real or complex vector space equipped with a norm ∥ · ∥. As in Section 11, a bounded linear functional on V is a bounded linear mapping from V into the real or complex numbers, as appropriate. The dual space V^{∗} of bounded linear functionals is thus the same as BL(V, R) or BL(V, C), and the dual norm ∥ · ∥_{∗} on V^{∗} is defined to be the corresponding operator norm, using the standard norm on R or C. In particular, V^{∗} is automatically complete with respect to the dual norm, as in the previous section.

Suppose that the norm ∥v∥ on V is associated to an inner product ⟨v, w⟩. Each w ∈ V determines a linear functional λ_{w} on V, defined by

λ_{w}(v) = ⟨v, w⟩. (14.1)

The Cauchy–Schwarz inequality implies that λ_{w} is a bounded linear functional on V, and that ∥λ_{w}∥_{∗} = ∥w∥. (14.2)

If V has finite dimension, then every linear functional on V is of the form λ_{w} for some w ∈ V. If V is a Hilbert space, then every bounded linear functional on V is of this form.

To see this, let λ be a bounded linear functional on V, which we may as well suppose is not identically equal to 0. Thus λ(z) = 1 for some z ∈ V, and we can put ρ = inf{∥z∥ : z ∈ V, λ(z) = 1}. (14.3)

Let {z_{j}}_{j=1}^{∞} be a sequence of elements of V such that λ(z_{j}) = 1 for each j and

lim_{j→∞} ∥z_{j}∥ = ρ. (14.4)

We would like to show that {z_{j}}_{j=1}^{∞} is a Cauchy sequence in V, and hence converges when V is a Hilbert space. Because the norm is associated to an inner product, we have the parallelogram law

∥\frac{v + w}{2} ∥^{2} + ∥\frac{v − w}{2} ∥^{2} = \frac{∥v∥^{2}}{2} + \frac{∥w∥^{2}}{2} (14.5)

for every v, w ∈ V. We want to apply this to terms of the sequence in order to show that ∥z_{j} − z_{l}∥ → 0 as j, l → ∞. (14.6)

For each j, l ≥ 1, λ(z_{j}) = λ(z_{l}) = 1, which implies that

λ(\frac{z_{j} + z_{l}}{2} ) = 1, (14.7)

and hence

∥\frac{z_{j} + z_{l}}{2} ∥ ≥ ρ. (14.8)

Using the parallelogram law, this implies in turn that

ρ^{2} + ∥\frac{z_{j} − z_{l}}{2} ∥^{2} ≤ \frac{∥z_{j}∥^{2}}{2} + \frac{∥z_{l}∥^{2}}{2} . (14.9)

It follows that {z_{j}}_{j=1}^{∞} is a Cauchy sequence, since ∥z_{j}∥, ∥z_{l}∥ → ρ as j, l → ∞.

Assuming that V is a Hilbert space, we get that {z_{j}}_{j=1}^{∞} converges to an element z of V. By construction, λ(z) = 1, and ∥z∥ = ρ is minimal among vectors with this property. If v ∈ V is in the kernel of λ, so that λ(v) = 0, then v is orthogonal to z, which is to say that

⟨v, z⟩ = 0. (14.10)

This is because λ(z + t v) = 1 (14.11)

for every t ∈ R or C, as appropriate, and hence

∥z + t v∥ ≥ ∥z∥ (14.12)

by minimality, which implies orthogonality, as in calculus. If w = z/∥z∥^{2}, then it follows that λ(v) = ⟨v, w⟩ = λ_{w}(v) (14.13)

for every v ∈ V, as desired.

15 ℓ^{1}(E)^{∗}

Let E be a nonempty set, and let h be a bounded real or complex-valued function on E. If f is a summable function on E, then f h is a summable function on E, and we can put

λ_{h}(f) = ∑_{x∈E} f(x) h(x). (15.1)

This defines a bounded linear functional on ℓ^{1}(E), since

|λ_{h}(f)| ≤ ∑_{x∈E} |f(x)| |h(x)| ≤ ∥h∥_{∞} ∥f∥_{1}. (15.2)

For each y ∈ E, let δ_{y} be the function on E defined by δ_{y}(x) = 0 when x ≠ y and δ_{y}(y) = 1. Thus ∥δ_{y}∥_{1} = 1 and

λ_{h}(δ_{y}) = h(y) (15.3)

for each y ∈ E, and one can use this to show that

∥λ_{h}∥_{1,∗} = ∥h∥_{∞}, (15.4)

where ∥ · ∥_{1,∗} denotes the dual norm with respect to the norm ∥f∥_{1} on ℓ^{1}(E).

Conversely, suppose that λ is a bounded linear functional on ℓ^{1}(E). For each y ∈ E, put h(y) = λ(δ_{y}). (15.5)

Thus |h(y)| ≤ ∥λ∥_{1,∗} (15.6)

for each y ∈ E, since ∥δ_{y}∥_{1} = 1. Equivalently, h is bounded on E, and

∥h∥_{∞} ≤ ∥λ∥_{1,∗}. (15.7)

It remains to show that λ = λ_{h} as linear functionals on ℓ^{1}(E).

More precisely, we want to show that for each f ∈ ℓ^{1}(E),

λ(f) = λ_{h}(f) (15.8)

If f(x) ≠ 0 for only finitely many x ∈ E, then f is a linear combination of finitely many δ_{y}’s, and this holds by linearity. Because these functions are dense in ℓ^{1}(E), the same conclusion holds for every f ∈ ℓ^{1}(E), by continuity.

16 c_{0}(E)^{∗}

Let E be a nonempty set, and let h be a summable real or complex-valued function on E. If f is a bounded function on E, then f h is a summable function on E, and we can put

λ_{h}(f) = ∑_{x∈E} f(x) h(x). (16.1)

This defines a bounded linear functional on ℓ^{∞}(E), since

|λ_{h}(f)| ≤ ∑_{x∈E} |f(x)| |h(x)| ≤ ∥h∥_{1} ∥f∥_{∞}. (16.2)

If we choose f so that ∥f∥_{∞} = 1 and

f(x) h(x) = |h(x)| (16.3)

for each x ∈ E, then λ_{h}(f) = ∥h∥_{1}, (16.4)

at it follows the dual norm of λ_{h} on ℓ^{∞}(E) is equal to ∥h∥_{1}.

We can also restrict λ_{h} to a bounded linear functional on c_{0}(E) with dual norm less than or equal to ∥h∥_{1}. If E_{1} ⊆ E has only finitely many elements, then we can choose a function f on E such that f(x) = 0 on E\E_{1}, ∥f∥_{∞} = 1, and (16.3) holds on E_{1}. Thus f ∈ c_{0}(E) and

λ_{h}(f) = ∑_{x∈E_{1}} |h(x)|, (16.5)

and one can use this to show that the dual norm of the restriction of λ_{h} to c_{0}(E) is still equal to ∥h∥_{1}, by taking the supremum over all finite subsets E_{1} of E.

Now let λ be any bounded linear functional on c_{0}(E). As in the previous section, we can define a function h on E by

h(y) = λ(δ_{y}). (16.6)

By construction,

λ(f) = ∑_{x∈E} f(x) h(x) (16.7)

when f(x) ≠ 0 for only finitely many x ∈ E. If E_{1} ⊆ E has only finitely many elements, then we can choose f as in the preceding paragraph to get that

∑_{x∈E_{1}} |h(x)| (16.8)

is less than or equal to the dual norm of λ on c_{0}(E). It follows that h is a summable function on E, and that ∥h∥_{1} is less than or equal to the dual norm of λ on c_{0}(E).

It remains to check that λ(f) = λ_{h}(f) (16.9)

for every f ∈ c_{0}(E). We already know that this holds when f(x) ≠ 0 for only finitely many x ∈ E, and it holds for every f ∈ c_{0}(E) by continuity, since these functions are dense in c_{0}(E).

17 Hölder’s inequality

Let 1 < p < ∞ be given, and let 1 < q < ∞ be the conjugate exponent, characterized by

\frac{1}{p} + \frac{1}{q} = 1. (17.1)

If a and b are nonnegative real numbers, then

a b ≤ \frac{a^{p}}{p} + \frac{b^{q}}{q} . (17.2)

This can be seen as a consequence of the convexity of the exponential function, and the p = q = 2 case is elementary, as in Section 7.

Suppose that f, h are real or complex-valued functions on a nonempty set E. As in the previous paragraph,

|f(x)| |h(x)| ≤ \frac{|f(x)|^{p}}{p} + \frac{|h(x)|^{q}}{q} (17.3)

for every x ∈ E. If f is p-summable and h is q-summable, then it follows that f h is summable on E, with

∑_{x∈E} |f(x)| |h(x)| ≤ \frac{1}{p} ∑_{x∈E} |f(x)|^{p} + \frac{1}{q} ∑_{x∈E} |h(x)|^{q}. (17.4)

Equivalently,

∥f h∥_{1} ≤ \frac{∥f∥_{p}^{p}}{p} + \frac{∥h∥_{q}^{q}}{q} . (17.5)

If ∥f∥_{p} = ∥h∥_{q} = 1, then we get that

∥f h∥_{1} ≤ 1. (17.6)

This implies for any f ∈ ℓ^{p}(E) and h ∈ ℓ^{q}(E) that

∥f h∥_{1} ≤ ∥f∥_{p} ∥h∥_{q}, (17.7)

by homogeneity. This is Hölder’s inequality.

The analogous statement also holds for p = 1 and q = 1, and was implicitly used in the previous two sections.

18 ℓ^{p}(E)^{∗}

Let 1 < p, q < ∞ be conjugate exponents again, and let h be a q-summable function on a nonempty set E. If f is a p-summable function on E, then f h is summable, and we can put

λ_{h}(f) = ∑_{x∈E} f(x) h(x). (18.1)

This defines a bounded linear functional on ℓ^{p}(E), since

|λ_{h}(f)| ≤ ∥h∥_{q} ∥f∥_{p}, (18.2)

by Hölder’s inequality.

If we choose f so that

f(x) h(x) = |h(x)|^{q} (18.3)

for every x ∈ E, and f(x) = 0 when h(x) = 0, then

|f(x)| = |h(x)|^{q−1}, (18.4)

and |f(x)|^{p} = |h(x)|^{q}, (18.5)

since p(q − 1) = q. In this case,

λ_{h}(f) = ∥h∥_{q}^{q} = ∥h∥_{q} ∥f∥_{p}, (18.6)

and it follows that the dual norm of λ_{h} on ℓ^{p}(E) is ∥h∥_{q}.

Now let λ be any bounded linear functional on ℓ^{p}(E), and let h be the function on E defined by h(y) = λ(δ_{y}), so that

λ(f) = ∑_{x∈E} f(x) h(x) (18.7)

when f(x) ≠ 0 for only finitely many x ∈ E. If E_{1} ⊆ E has only finitely many elements, then we can choose f so that f(x) = 0 when x ∈ E\E_{1} or h(x) = 0 and (18.3) holds for every x ∈ E_{1}, and we get that

λ_{h}(f) = ∑_{x∈E_{1}} |h(x)|^{q}. (18.8)

We also have that

∥f∥_{p} = ( ∑_{x∈E_{1}} |h(x)|^{q})^{1/p}, (18.9)

which implies that

( ∑_{x∈E_{1}} |h(x)|^{q})^{1/q} (18.10)

is bounded by the dual norm of λ on ℓ^{p}(E). It follows that h is q-summable, and that ∥h∥_{q} is less than or equal to the dual norm of λ on ℓ^{p}(E).

The remaining point is that λ(f) = λ_{h}(f) for every f ∈ ℓ^{p}(E). This holds by construction when f(x) ≠ 0 for only finitely many x ∈ E, and it is easy to see that these functions are dense in ℓ^{p}(E). Thus λ(f) = λ_{h}(f) for every f ∈ ℓ^{p}(E) by continuity.

19 Separability

Remember that a metric space is said to be separable if it contains a dense set with only finitely or countably many elements. For example, the real line is separable, because the rationals are countable and dense. Similarly, R^{n} and C^{n} are separable for each positive integer n.

Let V be a real or complex vector space equipped with a norm. If there is a set A ⊆ V with only finitely or countably many elements whose span is dense in V, then V is separable. In the real case, the set of linear combinations of elements of A with rational coefficients is a countable dense set in V. In the complex case, one can use coefficients with rational real and imaginary parts.

For example, ℓ^{p}(E) is separable when E is countable and 1 ≤ p < ∞, because the linear span of the functions δ_{y}, y ∈ E, is dense. The same argument applies to c_{0}(E) equipped with the supremum norm, and to ℓ^{p}(E) with 0 < p < 1, using the metric ∥f_{1} − f_{2}∥_{p}^{p}. The space of continuous functions on the unit interval is separable with respect to the supremum norm, because polynomials are dense.

One can show that the space of continuous functions on any compact metric space is separable with respect to the supremum norm. This uses the fact that continuous functions on compact metric spaces are automatically uniformly continuous.

20 An extension lemma

Let V be a real vector space, let W be a linear subspace of codimension 1 in V, and let z be an element of V \W. Thus every v ∈ V can be expressed in a unique way as v = w + t z, (20.1)

where w ∈ W and t ∈ R. If λ is a linear functional on W, then any extension of λ to a linear functional μ on V is uniquely determined by its value α ∈ R at z, since μ(w + t z) = λ(w) + t α. (20.2)

Any choice of α ∈ R defines an extension μ of λ in this way.

Suppose now that V is equipped with a norm ∥ · ∥, and that λ is a bounded linear functional with respect to the restriction of ∥ · ∥ to W. Thus there is an L ≥ 0 such that |λ(w)| ≤ L ∥w∥ (20.3)

for every w ∈ W. We would like to choose α ∈ R and hence μ so that

|μ(v)| ≤ L ∥v∥ (20.4)

for every v ∈ V, which is the same as

|λ(w) + t α| ≤ L ∥w + t z∥ (20.5) for every w ∈ W and t ∈ R. This holds by hypothesis when t = 0, and for t ≠ 0 it is enough to show that |λ(w) + α| ≤ L ∥w + z∥ (20.6) for every w ∈ W, since otherwise we can replace w by t w and then take out |t| from both sides by homogeneity.

Equivalently, we would like to choose α ∈ R such that −L ∥w + z∥ − λ(w) ≤ α ≤ L ∥w + z∥ − λ(w) (20.7) for every w ∈ W. To do this, it suffices to show that −L ∥w_{1} + z∥ − λ(w_{1}) ≤ L ∥w_{2} + z∥ − λ(w_{2}) (20.8) for every w_{1}, w_{2} ∈ W. By hypothesis, we know that λ(w_{2} − w_{1}) ≤ L ∥w_{2} − w_{1}∥ (20.9) for every w_{1}, w_{2} ∈ W, which implies that λ(w_{2} − w_{1}) ≤ L ∥w_{2} + z∥ + L ∥w_{1} + z∥, (20.10) because of the triangle inequality. This gives the desired estimate.

21 The Hahn–Banach theorem

Let V be a real vector space equipped with a norm ∥ · ∥, and let W be a linear subspace of V. The Hahn–Banach theorem asserts that every bounded linear functional λ on W has an extension to a bounded linear functional μ on V with the same norm.

If W has finite codimension in V, then one can apply the procedure described in the previous section finitely many times. If V is separable, then one can apply this procedure repeatedly to extend λ to a dense linear subspace of V, and then to all of V by continuity. Otherwise, there is an argument based on the axiom of choice.

As an application, it follows that for every v ∈ V there is a bounded linear functional μ on V with norm 1 such that μ(v) = ∥v∥. (21.1) To see this, one can start with the linear functional on the 1-dimensional subspace spanned by v defined by λ(r v) = r ∥v∥, r ∈ R. (21.2) This has norm 1, and a norm-1 extension to V has the desired properties.

Similarly, if W_{0} is a closed linear subspace of V and v ∈ V \W_{0}, then there is a bounded linear functional μ on V such that

μ(v) ≠ 0 (21.3)

and μ(w) = 0 (21.4)

for every w ∈ W_{0}. For it is easy to find a bounded linear functional on the span of W_{0} and v with these properties, which can then be extended to V.

22 Complex vector spaces

Let V be a complex vector space, which is then also a real vector space, since we can simply forget about multiplication by i. If λ is a linear functional on V as a complex vector space, then the real part of λ is a linear functional on V as a real vector space. The imaginary part of λ can be recovered from the real part, by the formula Im λ(v) = i Re λ(−i v). (22.1)

Similarly, any linear functional on V as a real vector space can be expressed as the real part of a linear functional on V as a complex vector space.

Suppose that V is equipped with a norm ∥ · ∥, which is also a norm on V as a real vector space. If λ is a bounded linear functional on V as a complex vector space, then one can show that the norm of λ is equal to the norm of the real part of λ as a bounded linear functional on V as a real vector space. This uses the fact that the norm of any v ∈ V is equal to the norm of c v for every complex number c with |c| = 1. Also, the modulus of a complex number z is the same as the maximum of the real parts of the complex numbers c z with c ∈ C and |c| = 1. Every bounded linear functional on V as a real vector space is the real part of a bounded linear functional on V as a complex vector space, as in the previous paragraph.

This permits questions about linear functionals on complex vector spaces to be reduced to questions about linear functionals on real vector spaces. In particular, the Hahn–Banach theorem also applies to complex vector spaces.

23 The weak^{∗} topology

Let V be a real or complex vector space equipped with a norm ∥v∥, and let V^{∗} be the dual space of bounded linear functionals on V with the dual norm ∥λ∥_{∗}. As usual, the dual norm determines a metric on V^{∗}, and hence a topology on V^{∗}. We are also interested in another topology on V^{∗}, known as the weak^{∗} topology.

A set U ⊆ V^{∗} is an open set in the weak^{∗} topology if for every λ ∈ U there are finitely many vectors v_{1}, …, v_{n} ∈ V and positive real numbers r_{1}, …, r_{n} such that {μ ∈ V^{∗} : |μ(v_{j}) − λ(v_{j})| < r_{j}, 1 ≤ j ≤ n} ⊆ U. (23.1)

Equivalently, {μ ∈ V^{∗} : |μ(v_{j}) − λ(v_{j})| < r_{j}, 1 ≤ j ≤ n} (23.2)

is an open set in V^{∗} with respect to the weak^{∗} topology for every λ ∈ V^{∗}, v_{1}, …, v_{n} ∈ V, and r_{1}, …, r_{n} > 0, and these sets form a base for the weak^{∗}- topology.

If U ⊆ V^{∗} is an open set with respect to the weak^{∗} topology, then U is also an open set with respect to the topology determined by the dual norm. However, open balls in V^{∗} with respect to the dual norm are not open sets in the weak^{∗} topology, unless V is finite-dimensional.

For each v ∈ V, λ ↦ λ(v) (23.3)

defines a linear functional on V^{∗}. More precisely, this is a bounded linear functional on V^{∗} with respect to the dual norm, since

|λ(v)| ≤ ∥λ∥_{∗} ∥v∥, (23.4)

by definition of the dual norm. The norm of this linear functional on V^{∗} is actually equal to ∥v∥, by the Hahn–Banach theorem. These linear functionals are also continuous on V^{∗} with respect to the weak^{∗} topology, by definition of the weak^{∗} topology. The weak^{∗} topology can be described as the weakest topology on V^{∗} in which these linear functionals are continuous.

24 Seminorms

A nonnegative real-valued function N(w) on a real or complex vector space W is said to be a seminorm if

N(t w) = |t| N(w) (24.1)

for every w ∈ W and t ∈ R or C, as appropriate, and

N(w + z) ≤ N(w) + N(z) (24.2)

for every w, z ∈ W. These are the same conditions as for a norm, except that N(w) = 0 may occur when w ≠ 0.

Suppose that N is a collection of seminorms on W which satisfies the positivity condition that for every w ∈ W with w ≠ 0,

N(w) > 0 (24.3)

for some N ∈ N. As in the case of a single norm, there is a nice topology on W determined by N, Specifically, U ⊆ W is an open set in this topology if for every u ∈ U there are finitely many seminorms N_{1}, …, N_{l} ∈ N and positive real numbers r_{1}, …, r_{l} such that

{w ∈ W : N_{j}(w − u) < r_{j}, 1 ≤ j ≤ l} ⊆ U. (24.4)

Equivalently, {w ∈ W : N_{j}(w − u) < r_{j}, 1 ≤ j ≤ l} (24.5)

is an open set in W with respect to this topology for every u ∈ W, N_{1}, …, N_{l} in N, and r_{1}, …, r_{l} > 0, and these sets form a base for the topology. The positivity condition ensures that this topology is Hausdorff.

For example, if μ is a linear functional on W, then

N(w) = |μ(w)| (24.6)

is a seminorm on W. The weak^{∗} topology on the dual V^{∗} of a vector space V corresponds to the family of seminorms

N_{v}(λ) = |λ(v)|, v ∈ V. (24.7)

If N consists of only finitely many seminorms N_{1}, …, N_{k} on W, then their maximum N(w) = max(N_{1}(w), …, N_{k}(w)) (24.8)

is a norm on W, and the topology on W corresponding to N is the same as the usual one determined by N.

If N consists of an infinite sequence N_{1}, N_{2}, … of seminorms on W, then the corresponding topology on W can be described by a metric. For each j ≥ 1, one can check that ρ_{j}(u, w) = min(N_{j}(u − w), 1/j) (24.9)

is a semimetric on W. This means that ρ_{j}(u, w) satisfies the same conditions as a metric, except that ρ_{j}(u, w) = 0 can occur when u ≠ w. One can also check that ρ(u, w) = max{ρ_{j}(u, w) : j ≥ 1} (24.10)

is a metric on W, and that the topology on W determined by this metric is the same as the topology corresponding to the sequence of seminorms as before.

25 The weak^{∗} topology, 2

Let V be a real or complex vector space with a norm ∥v∥, and let V^{∗} be the dual space of bounded linear functionals on V with the dual norm ∥λ∥_{∗}. Also let B^{∗} = {λ ∈ V^{∗} : ∥λ∥_{∗} ≤ 1} (25.1)

be the closed unit ball in V^{∗}.

As in the previous section, the weak^{∗} topology on V^{∗} is the same as the topology corresponding to the family of seminorms

N_{v}(λ) = |λ(v)|, v ∈ V. (25.2)

For A ⊆ V, consider the family of seminorms N_{A} on V^{∗} consisting of N_{v} with v ∈ A. If the linear span of A is dense in V, then the topology on B^{∗} induced by the weak^{∗} topology on V^{∗} is the same as the topology induced by the one corresponding to N_{A}. This is easy to see, and it works just as well for any subset of V^{∗} which is bounded with respect to the dual norm.

If V is separable, then we can take A to be a countable set. As in the previous section, the topology corresponding to N_{A} is then determined by a metric. Thus the topology on B^{∗} or other bounded subsets of V^{∗} induced by the weak^{∗} topology can be described by a metric in this case.

Technically, the term “weak^{∗} topology” on V^{∗} should perhaps only be used when V is complete. If V is not complete, then V can be isometrically embedded as a dense linear subspace of a Banach space. Although V and its completion have the same dual space, the additional elements of the completion can affect the weak^{∗} topology on the dual. However, this does not matter for the topology on bounded subsets of the dual induced by the weak^{∗} topology, by the earlier remarks.

26 The Banach–Alaoglu theorem

Let V be a real or complex vector space equipped with a norm ∥v∥, and let V^{∗} be the dual space with dual norm ∥λ∥_{∗}. The Banach–Alaoglu theorem states that the closed unit ball B^{∗} in V^{∗} is compact with respect to the weak^{∗} topology.

As usual, each v ∈ V determines a linear functional

λ ↦ λ(v) (26.1)

on V^{∗}. This defines a mapping from V^{∗} into the Cartesian product of copies of R or C, as appropriate, parameterized by v ∈ V. One can check that this mapping is a homeomorphism from V^{∗} onto its image in the Cartesian product with respect to the topology induced by the product topology, because of the way that the weak^{∗} topology is defined. One can also check that the image of B^{∗} under this mapping is a closed set with respect to the product topology. The image of B^{∗} is also contained in the product of closed intervals or disks, as appropriate, which implies compactness.

If V is separable, then one can argue more directly that every sequence {λ_{j}}_{j=1}^{∞} in B^{∗} has a subsequence that converges to an element of B^{∗} in the weak^{∗} topology. For each v ∈ V, one can use the compactness of closed and bounded subsets of R or C to get a subsequence {λ_{j_{l}}}_{l=1}^{∞} of {λ_{j}}_{j=1}^{∞} such that {λ_{j_{l}}(v)}_{j=1}^{∞} converges. By standard arguments, this can be refined to show that there is a subsequence which converges pointwise on a countable subset of V. Hence there is a subsequence that converges on a dense set in V when V is separable. Using the fact that these linear functionals all have norm less than or equal to 1, one can show that this subsequence actually converges pointwise everywhere on V, and that the limit is also a linear functional with norm less than or equal to 1.

27 The weak topology

Let V be a real or complex vector space equipped with a norm ∥v∥. As in Section 24 each bounded linear functional λ on V determines a seminorm on V, given by N_{λ}(v) = |λ(v)|. (27.1)

The topology on V defined by the family of seminorms N_{λ}, λ ∈ V^{∗}, is called the weak topology. Note that for each v ∈ V there is a λ ∈ V^{∗} such that λ(v) ≠ 0 and hence N_{λ}(v) > 0, by the Hahn–Banach theorem, so that this family of seminorms satisfies the usual positivity condition.

If U ⊆ V is an open set with respect to the weak topology, then U is also an open set with respect to the topology associated to the norm. However, an open ball in V is not an open set with respect to the weak topology, unless V has finite dimension. If W is a linear subspace of V which is closed in the norm topology, then W is also closed in the weak topology, by the Hahn–Banach theorem. Each bounded linear functional on V is also continuous with respect to the weak topology, and the weak topology is the weakest topology on V with this property.

The weak topology can also be defined on the dual V^{∗} of V, using bounded linear functionals on V^{∗} with respect to the dual norm. As in Section 23,

λ ↦ λ(v) (27.2)

defines a bounded linear functional on V^{∗} for each v ∈ V, which implies that every open set in V^{∗} with respect to the weak^{∗} topology is also an open set in the weak topology on V^{∗}. If every bounded linear functional on V^{∗} is given by evaluation at an element of V in this way, then V is said to be reflexive. Thus the weak and weak^{∗} topologies on V^{∗} are the same when V is reflexive.

Hilbert spaces are automatically reflexive, because of the way that they can be identified with their dual spaces. If E is a nonempty set and 1 < p < ∞, then ℓ^{p}(E) is also reflexive. As in Section 18, the dual of ℓ^{p}(E) can be identified with ℓ^{q}(E), where q is the conjugate exponent corresponding to p. For the same reasons, the dual of ℓ^{q}(E) can be identified with ℓ^{p}(E). As in Sections 15 and 16, the dual of c_{0}(E) can be identified with ℓ^{1}(E) and the dual of ℓ^{1}(E) can be identified with ℓ^{∞}(E), which is not the same as c_{0}(E) when E has infinitely many elements.

28 Seminorms, 2

Let W be a real or complex vector space, and let N be a family of seminorms on W that satisfies the positivity condition that for each w ∈ W there is an N ∈ N such that N(w) > 0. (28.1)

As in Section 24, this leads to a nice topology on W. Let us say that a linear functional ψ on W is bounded with respect to N if there are finitely many seminorms N_{1}, …, N_{l} ∈ N and A ≥ 0 such that

|ψ(w)| ≤ A max(N_{1}(w), …, N_{l}(w)) (28.2)

for each w ∈ W. This implies that

|ψ(u) − ψ(w)| ≤ A max(N_{1}(u − w), …, N_{l}(u − w)) (28.3)

for every u, w ∈ W, and it is easy to see that ψ is therefore continuous with respect to the topology on W corresponding to N.

Conversely, suppose that ψ is a linear functional on W which is continuous with respect to the topology corresponding to N. This implies that

U = {w ∈ w : |ψ(w)| < 1} (28.4)

is an open set in this topology. Since 0 ∈ U, this means that there are finitely many seminorms N_{1}, …, N_{l} ∈ N and positive real numbers r_{1}, …, r_{l} > 0 such that {w ∈ w : N_{j}(w) < r_{j}, 1 ≤ j ≤ l} ⊆ U, (28.5)

by definition of the topology on W corresponding to N. Equivalently, |ψ(w)| < 1 when N_{1}(w) < r_{1}, …, N_{l}(w) < r_{l}, (28.6)

which implies that

|ψ(w)| ≤ max(r_{1}^{−1} N_{1}(w), …, r_{l}^{−1} N_{l}(w)), (28.7)

and hence that ψ is bounded with respect to N.

For example, let Φ be a collection of linear functionals on W with the property that for each w ∈ W there is a ϕ ∈ Φ such that ϕ(w) ≠ 0. Let N_{Φ} be the family of seminorms N_{ϕ}(w) = |ϕ(w)| (28.8)

on W, ϕ ∈ Φ, which satisfies the positivity condition by hypothesis. Each ϕ ∈ Φ is a continuous linear functional on W with respect to the topology corresponding to N_{Φ}, as well as any finite linear combination of elements of Φ.

Conversely, any linear functional ψ on W which is continuous with respect to the topology associated to N_{Φ} can be expressed as a linear combination of finitely many elements of Φ. For if ψ is continuous, then it is bounded with respect to N_{Φ}, and there are ϕ_{1}, …, ϕ_{l} ∈ Φ and A ≥ 0 such that

|ψ(w)| ≤ A max(|ϕ_{1}(w)|, …, |ϕ_{l}(w)|) (28.9)

for each w ∈ W. In particular, ψ(w) = 0 when

ϕ_{1}(w) = ⋯ = ϕ_{l}(w) = 0. (28.10)

It follows from standard arguments in linear algebra that ψ is a linear combination of ϕ_{1}, …, ϕ_{l}, as desired.

29 Convergent sequences

Let (M, d(·, ·)) be a metric space. By definition, a sequence {x_{j}}_{j=1}^{∞} of elements of M converges to x ∈ M if and only if

lim_{j→∞} d(x_{j}, x) = 0 (29.1)

as a sequence of real numbers. In particular, if W is a vector space with a norm ∥ · ∥_{W}, then a sequence of vectors {w_{j}}_{j=1}^{∞} converges to w ∈ W with respect to the norm if and only if

lim_{j→∞} ∥w_{j} − w∥_{W} = 0. (29.2)

If instead the topology on W is determined by a family N of seminorms on W, then one can check that {w_{j}}_{j=1}^{∞} converges to w if and only if

lim_{j→∞} N(w_{j} − w) = 0 (29.3)

for each N ∈ N.

Let V be a real or complex vector space equipped with a norm ∥ · ∥, and let V^{∗} be the dual space of bounded linear functionals on V equipped with the dual norm ∥ · ∥_{∗}. A sequence {v_{j}}_{j=1}^{∞} of vectors in V converges to v ∈ V in the weak topology if and only if

lim_{j→∞} λ(v_{j}) = λ(v) (29.4)

as a sequence of real or complex numbers for each λ ∈ V^{∗}. Similarly, a sequence {λ_{j}}_{j=1}^{∞} of bounded linear functionals on V converges to λ ∈ V^{∗} in the weak^{∗} topology if and only if

lim_{j→∞} λ_{j}(v) = λ(v) (29.5)

as a sequence of real or complex numbers for each v ∈ V.

In a metric space M, the topology is determined by convergence of sequences, because closed subsets of M can be characterized in terms of convergent sequences. This also works in topological spaces which satisfy the first countability condition, for which there is a countable local base for the topology at every point, but it does not work in arbitrary topological spaces. In some circumstances, the induced topology on interesting subsets of a topological space X may satisfy the first countability condition, even if X does not enjoy this property itself. For example, this is the case for bounded subsets of the dual V^{∗} of a separable vector space V with a norm, with respect to the weak^{∗} topology on V^{∗}.

Suppose that {λ_{j}}_{j=1}^{∞} is a sequence of bounded linear functionals on a vector space V with a norm which is uniformly bounded in the sense that

∥λ_{j}∥_{∗} ≤ L (29.6)

for some L ≥ 0 and each j ≥ 1. If {λ_{j}(v)}_{j=1}^{∞} converges as a sequence of real or complex numbers for a set of v ∈ V whose linear span is dense in V, then one can show that {λ_{j}(v)}_{j=1}^{∞} is a Cauchy sequence in R or C for every v ∈ V, and hence converges. If we put

λ(v) = lim_{j→∞} λ_{j}(v) (29.7)

for each v ∈ V, then it is easy to see that λ is a bounded linear functional on V, with ∥λ∥_{∗} ≤ L, (29.8)

and {λ_{j}}_{j=1}^{∞} converges to λ in the weak^{∗} topology.

30 Uniform boundedness

Let V be a real or complex vector space with a norm ∥·∥, and let V^{∗} be the dual space of bounded linear functionals on V with the dual norm ∥ · ∥_{∗}. Suppose that {λ_{j}}_{j=1}^{∞} is a sequence in V^{∗} such that {λ_{j}(v)}_{j=1}^{∞} converges in R or C for every v ∈ V, which implies that {λ_{j}(v)}_{j=1}^{∞} is bounded for each v ∈ V. If V is complete, then a theorem of Banach and Steinhaus implies that {λ_{j}}_{j=1}^{∞} is bounded in V^{∗}, so that ∥λ_{j}∥_{∗} ≤ L (30.1)

for some L ≥ 0 and every j ≥ 1.

To see this, put

(30.2) A(n) = {v ∈ V : |λ_{j}(v)| ≤ n for every j ≥ 1}

for each positive integer n. Because λ_{j} is bounded and hence continuous for every j, A(n) is a closed set in V for each n, while

⋃_{n=1}^{∞} A(n) = V, (30.3)

by hypothesis. Hence A(n) contains a nonempty open set in V for some n, by the Baire category theorem, since V is complete. Thus the λ_{j}’s are uniformly bounded on a nonempty open set in V, and one can use this and linearity to show that their dual norms are bounded.

Similarly, if {v_{j}}_{j=1}^{∞} is a sequence in V such that {λ(v_{j})}_{j=1}^{∞} converges in R or C for every λ ∈ V^{∗}, then one can show that the v_{j}’s have bounded norm. As in the previous case, it is actually sufficient to know that {λ(v_{j})}_{j=1}^{∞} is bounded for each λ ∈ V^{∗}. More precisely, one can first use the Baire category theorem to show that there is an L′ ≥ 0 such that

|λ(v_{j})| ≤ L′ ∥λ∥_{∗} (30.4)

for every λ ∈ V^{∗} and j ≥ 1, as before, and then the Hahn–Banach theorem implies that ∥v_{j}∥ ≤ L′ (30.5)

for each j ≥ 1, as desired.

31 Examples

Let E be a nonempty set, and let 1 ≤ p, q ≤ ∞ be conjugate exponents. As in Sections 15, 16, and 18, the dual of ℓ^{p}(E) can be identified with ℓ^{q}(E) when 1 ≤ p < ∞, and the dual of c_{0}(E) with the ℓ^{∞} norm can be identified with ℓ^{1}(E). Hence the weak^{∗} topology can be defined on ℓ^{q}(E) for 1 ≤ q ≤ ∞ using these identifications.

By definition, a sequence {f_{j}}_{j=1}^{∞} of functions on E in ℓ^{q}(E) converges to another function f ∈ ℓ^{q}(E) in the weak^{∗} topology if

lim_{j→∞} ∑_{x∈E} f_{j}(x) h(x) = ∑_{x∈E} f(x) h(x) (31.1)

for every h ∈ ℓ^{p}(E) when 1 ≤ p < ∞, and every h ∈ c_{0}(E) when p = ∞. This implies that

lim_{j→∞} f_{j}(x) = f(x) (31.2)

for every x ∈ E, by taking h to be equal to 1 at x and 0 elsewhere. Moreover, weak^{∗} convergence implies that there is an L ≥ 0 such that

∥f_{j}∥_{q} ≤ L, (31.3)

as in the previous section.

Conversely, (31.2) and (31.3) imply that f ∈ ℓ^{q}(E), with

∥f∥_{q} ≤ L. (31.4)

This is a straightforward consequence of the definitions when q = ∞, while for 1 ≤ q < ∞ one can observe first that

( ∑_{x∈E_{1}} |f(x)|^{q})^{1/q} = lim_{j→∞} ( ∑_{x∈E_{1}} |f_{j}(x)|^{q})^{1/q} ≤ L (31.5)

for every set E_{1} ⊆ E with only finitely many elements. Of course, (31.2) implies (31.1) when h(x) ≠ 0 for only finitely many x ∈ E. One can extend this to all h ∈ ℓ^{p}(E) when 1 ≤ p < ∞ and to all h ∈ c_{0}(E) when p = ∞ by approximation, using the uniform bound (31.3).

Similarly, a sequence {f_{j}}_{j=1}^{∞} of summable functions on E converges to a summable function f on E in the weak topology on ℓ^{1}(E) if (31.1) holds for every h ∈ ℓ^{∞}(E). In particular, this implies that

lim_{j→∞} ∑_{x∈E} f_{j}(x) = ∑_{x∈E} f(x). (31.6)

For instance, if {f_{j}}_{j=1}^{∞} is a sequence of nonnegative real-valued functions on E which converges weakly to 0, then (31.6) implies that

lim_{j→∞} ∥f_{j}∥_{1} = 0. (31.7)

However, it is easy to give examples of sequences of nonnegative real-valued summable functions that converge to 0 in the weak^{∗} topology and not in the ℓ^{1} norm.

32 An embedding

Let V be a vector space with norm ∥v∥, and let V^{∗} be its dual space with dual norm ∥λ∥_{∗}. Also let X be the closed unit ball in V^{∗}, consisting of λ ∈ V^{∗} with ∥λ∥_{∗} ≤ 1, equipped with the weak^{∗} topology. Thus X is a compact Hausdorff space, by the Banach–Alaoglu theorem.

As usual, each v ∈ V determines a bounded linear functional η_{v} on V^{∗}, defined by η_{v}(λ) = λ(v). (32.1)

The restriction of η_{v} to the unit ball of V^{∗} defines a continuous function on X for each v ∈ V, so that we get a linear mapping from V into the space C(X) of continuous real or complex-valued functions on X, as appropriate. This mapping is an isometry, since for each v ∈ V,

∥η_{v}∥_{∞} = ∥η_{v}∥_{∗} = ∥v∥. (32.2)

Here ∥η∥_{∞} denotes the supremum norm of a continuous function η on X.

If V is separable, then the topology on X can be described by a metric, as in Section 25. In particular, X is separable, since it is compact. If E is a countable dense set in X, then there is a natural isometric embedding of C(X) into ℓ^{∞}(E), defined by restricting continuous functions on X to E. Hence we get an isometric embedding of C(X) into ℓ^{∞}(E) too.

33 Induced mappings

Let X, Y be topological spaces, and let ϕ : X → Y be a continuous mapping between them. If f is a continuous function on Y, then

Φ(f) = f ∘ ϕ (33.1)

is a continuous function on X, and Φ determines a linear mapping from C_{b}(Y) into C_{b}(X). Moreover, ∥Φ(f)∥_{X,∞} ≤ ∥f∥_{Y,∞}, (33.2)

where the subscripts indicate on which space the supremum norm is defined, so that Φ : C_{b}(Y) → C_{b}(X) is a bounded linear operator with norm ≤ 1. It is easy to see that Φ actually has operator norm equal to 1, since Φ takes constant functions on Y to constant functions on X.

If ϕ(X) is dense in Y, then

∥Φ(f)∥_{X,∞} = ∥f∥_{Y,∞} (33.3)

for each bounded continuous function f on Y. Thus Φ is an isometric embedding of C_{b}(Y) into C_{b}(X) in this case. For example, one can apply this to the coordinate projections from the unit square [0, 1] × [0, 1] onto the unit interval [0, 1] to get isometric embeddings of C([0, 1]) into C([0, 1] × [0, 1]). One can also apply this to a continuous mapping ϕ from the unit interval [0, 1] onto the unit square [0, 1] × [0, 1], which is to say a space-filling curve, to get an isometric embedding of C([0, 1] × [0, 1]) into C([0, 1]).

There are sometimes linear operators which send continuous functions on a closed set A ⊆ X to continuous extensions on X, which can lead to embeddings of C_{b}(A) into C_{b}(X). For example, if X is the unit interval [0, 1], then a continuous function f on a closed set A ⊆ [0, 1] can be extended to a continuous function on [0, 1] which is linear on the intervals in [0, 1]\A. If A does not contain 0 or 1, then the extension can be taken to be constant on the corresponding interval in the complement. This leads to an isometric linear embedding of C(A) into C(0, 1).

In particular, this applies to the middle-thirds Cantor set. If X is a compact metric space, then there is a continuous mapping from the Cantor set onto X. It follows that there is an isometric linear embedding of C(X) into C(A), and hence into C([0, 1]). Using also the embedding described in the previous section, any separable Banach space can be isometrically embedded into C([0, 1]).

34 Continuity of norms

Let V be a real or complex vector space equipped with a norm ∥v∥. By the triangle inequality, ∥v∥ ≤ ∥w∥ + ∥v − w∥ (34.1)

and ∥w∥ ≤ ∥v∥ + ∥v − w∥ (34.2)

for each v, w ∈ V, and therefore

|∥v∥ − ∥w∥| ≤ ∥v − w∥ (34.3)

for every v, w ∈ V. This implies that ∥v∥ is a continuous function on V, with respect to the topology determined by the norm. Similarly, if N is a family of seminorms on V, then each seminorm N ∈ N is continuous with respect to the topology determined by N.

It is easy to show that any norm on R^{n} or C^{n} is bounded by a constant multiple of the standard norm, by expressing vectors as linear combinations of the standard basis and applying the triangle inequality. Hence the norm is continuous with respect to the standard topology on R^{n} or C^{n}. Because the unit sphere with respect to the standard norm is compact, any continuous function on the sphere attains its minimum, which is positive in the case of another norm. It follows that any norm on R^{n} or C^{n} is also bounded from below by a positive multiple of the standard norm, so that they are equivalent. If V is a finite-dimensional real or complex vector space, then V is isomorphic to R^{n} or C^{n} for some n, and the preceding remarks imply that any two norms on V are equivalent.

If V is an infinite-dimensional vector space equipped with a norm ∥v∥, then ∥v∥ is not a continuous function with respect to the weak topology, because open balls are not open subsets of V in the weak topology. However, closed balls are closed sets in the weak topology, which means that ∥v∥ is lower semicontinuous with respect to the weak topology. Similarly, closed balls in the dual space V^{∗} relative to the dual norm ∥λ∥_{∗} are closed sets with respect to the weak^{∗} topology, so that the dual norm is lower semicontinuous with respect to the weak^{∗} topology.

Every linear functional on R^{n} or C^{n} is bounded relative to the standard norm, as one can see using the standard basis again. Thus every linear functional on a finite-dimensional vector space V is bounded relative to any norm on V, by the earlier remarks, and the weak topology on V is the same as the topology determined by the norm. In this case, the dual space V^{∗} has the same finite dimension as V, and the weak^{∗} topology on V^{∗} is the same as the topology determined by the dual norm. Of course, finite-dimensional spaces are automatically reflexive.

35 Infinite series

Let V be a real or complex vector space equipped with a norm ∥v∥_{V}. An infinite series

∑_{j=1}^{∞} v_{j} (35.1)

with terms in V is said to converge if the corresponding sequence of partial sums

∑_{j=1}^{n} v_{j} (35.2)

converges in V. In this case, the value of the sum (35.1) is defined to be the limit of the sequence of partial sums.

Similarly, (35.1) is said to converge absolutely if

∑_{j=1}^{∞} ∥v_{j}∥ (35.3)

converges in the usual sense as an infinite series of nonnegative real numbers. As in the case of infinite series of real or complex numbers, absolute convergence of an infinite series in V implies that the corresponding sequence of partial sums is a Cauchy sequence. If V is complete, then absolute convergence of a series in V implies that the series converges in V.

Conversely, if every absolutely convergent series in V converges, then V is complete. For let {w_{j}}_{j=1}^{∞} be a Cauchy sequence in V, which we would like to show converges. By the definition of a Cauchy sequence, there is a subsequence {w_{j_{l}}}_{l=1}^{∞} such that ∥w_{j_{l+1}} − w_{j_{l}}∥ ≤ 2^{−l} (35.4)

for each l ≥ 1. Thus

∑_{l=1}^{∞}(w_{j_{l+1}} − w_{j_{l}}) (35.5)

converges absolutely, and so it also converges in V by hypothesis. Of course,

∑_{l=1}^{n}(w_{j_{l+1}} − w_{j_{l}}) = w_{n} − w_{1} (35.6)

for each n, so that convergence of the series implies that

lim_{n→∞} w_{j_{n}} = w_{1} + ∑_{l=1}^{∞}(w_{j_{l+1}} − w_{j_{l}}). (35.7)

In any metric space, a Cauchy sequence converges as soon as any of its subsequences converge, and to the same limit. It follows that {w_{j}}_{j=1}^{∞} also converges in V, as desired.

Convergence of infinite series in a vector space can also be defined when the topology on the vector space is determined by a family of seminorms, in terms of the convergence of the corresponding sequence of partial sums. In particular, this applies to convergence with respect to the weak and weak^{∗} topologies.

36 Some examples

Let us consider vector spaces of real or complex-valued functions on the set Z_{+} of positive integers. For each j ≥ 1, let δ_{j} be the function on Z_{+} equal to 1 at j and 0 elsewhere, as usual. If f is any function on Z_{+}, then

∑_{j=1}^{∞} f(j) δ_{j} (36.1)

converges to f pointwise on Z_{+}.

If f ∈ ℓ^{p}(Z_{+}), 1 ≤ p < ∞, then (36.1) converges to f in the ℓ^{p} norm. If f ∈ c_{0}(Z_{+}), then (36.1) converges to f in the ℓ^{∞}. Conversely, if f ∈ ℓ^{∞}(Z_{+}) and (36.1) converges to f in the ℓ^{∞} norm, then f ∈ c_{0}(Z_{+}). For any f ∈ ℓ^{∞}(Z_{+}), (36.1) converges to f in the weak^{∗} topology, where ℓ^{∞}(Z_{+}) is identified with the dual of ℓ^{1}(Z_{+}).

If f ∈ ℓ^{1}(Z_{+}), then (36.1) converges absolutely in ℓ^{1}(E). This does not work when p > 1, but there is a version of absolute convergence in the weak topology when 1 < p < ∞, and in the weak^{∗} topology when p = ∞. More precisely, let 1 ≤ p, q ≤ ∞ be conjugate exponents, and put

λ_{h}(f) = ∑_{x∈Z_{+}} f(x) h(x) (36.2)

when f ∈ ℓ^{p}(Z_{+}) and h ∈ ℓ^{q}(Z_{+}), which makes sense by Hölder’s inequality. Thus

∑_{j=1}^{∞} |λ_{h}(f(j) δ_{j})| = ∑_{j=1}^{∞} |f(j)| |h(j)| ≤ ∥f∥_{p} ∥h∥_{q}, (36.3)

and in particular the series on the left converges absolutely when f ∈ ℓ^{p}(Z_{+}) and h ∈ ℓ^{q}(Z_{+}).

If f ∈ ℓ^{p}(Z_{+}) and 0 < p < 1, then the partial sums of (36.1) converge to f with respect to the usual metric ∥f_{1} − f_{2}∥_{p}^{p}. The appropriate version of absolute convergence for a series of functions ∑_{j=1}^{∞} a_{j} in ℓ^{p}(E) on any set E when 0 < p < 1 is that

∑_{j=1}^{∞} ∥a_{j}∥_{p}^{p} (36.4)

converges as a series of nonnegative real numbers. This is satisfied by (36.1) when f ∈ ℓ^{p}(Z_{+}), as in the p = 1 case.

37 Quotient spaces

Let V be a real or complex vector space, and let W be a linear subspace of V. The quotient space V/W is defined by identifying v, v′ ∈ V when

v − v′ ∈ W. (37.1)

More precisely, this defines an equivalence relation on V, and the elements of V/W represent equivalence classes associated to this equivalence relation. The operations of addition and scalar multiplication on V lead to similar operations on V/W, so that V/W is also a real or complex vector space, as appropriate. By construction, there is a quotient map

q : V → V/W, (37.2)

that assigns to each v ∈ V the element q(v) of V/W corresponding to the equivalence class in V containing v, and which is a linear mapping from V onto V/W with kernel W.

Let ∥v∥_{V} be a norm on V. For each z ∈ V/W, put

∥z∥_{V/W} = inf{∥v∥_{V} : v ∈ V, q(v) = z}. (37.3)

It is easy to check that this defines a seminorm on V/W which is a norm exactly when W is a closed linear subspace of V with respect to the topology determined by ∥ · ∥_{V}.

If V is complete with respect to ∥ · ∥_{V} and W is a closed subspace of V, then V/W is complete with respect to the quotient norm. To see this, it suffices to show that any absolutely convergent series

∑_{j=1}^{∞} z_{j} (37.4)

of elements of V/W converges in V/W, as in Section 35. For each j ≥ 1, let v_{j} be an element of V such that q(v_{j}) = z_{j} and

∥v_{j}∥_{V} < ∥z_{j}∥_{V/W} + 2^{−j}. (37.5)

Thus

∑_{j=1}^{∞} v_{j} (37.6)

converges absolutely in V, and hence converges in V by completeness. This implies that (37.4) converges in V/W, with

∑_{j=1}^{∞} z_{j} = q( ∑_{j=1}^{∞} v_{j}). (37.7)

If V has finite dimension dim V, then W and V/W have finite dimensions as well, and (37.8) dim V = dim W + dim V/W.

For if v_{1}, …, v_{n} is a basis for V such that v_{1}, …, v_{l} is a basis for W, where

l = dim W ≤ dim V = n, (37.9)

then one can check that q(v_{l+1}), …, q(v_{n}) is a basis for V/W. As an extension of this to infinite dimensions, one can say that if V is separable, then V/W is separable, because q maps a countable dense set in V to a countable dense set in V/W.

38 Quotients and duality

Let V be a real or complex vector space equipped with a norm ∥ · ∥_{V}, let W be a closed linear subspace of V, and let V/W be the corresponding quotient space with quotient norm ∥ · ∥_{V/W}. Consider also the linear subspace of the dual space V^{∗} defined by

W^{⊥} = {λ ∈ V^{∗} : λ(w) = 0}. (38.1)

Note that W^{⊥} is closed with respect to the topology on V^{∗} determined by the dual norm, and even with respect to the weak^{∗} topology.

If μ is a bounded linear functional on V/W, then

λ = μ ∘ q (38.2)

is a bounded linear functional on V. By construction, λ ∈ W^{⊥}, and one can check that every λ ∈ W^{⊥} corresponds to a μ ∈ (V/W)^{⊥} in this way. One can also check that the dual norm of λ on V is equal to the dual norm of μ on V/W. This leads to a natural isometric equivalence

W^{⊥} ≅ (V/W)^{∗}. (38.3)

Similarly, if ϕ is a bounded linear functional on V, then the restriction of ϕ to W defines a bounded linear functional ψ on W whose dual norm is less than or equal to the dual norm of ϕ. Every bounded linear functional ψ on W is the restriction of a bounded linear functional ϕ on V to W with the same norm, by the Hahn–Banach theorem. Of course, the restriction of ϕ to W is 0 exactly when ϕ ∈ W^{⊥}. This leads to another natural isometric equivalence

W^{∗} ≅ V^{∗}/W^{⊥}. (38.4)

39 Dual mappings

Let V, W be vector spaces, both real or both complex, and equipped with norms ∥v∥_{V}, ∥w∥_{W}. If T : V → W is a bounded linear mapping, then the operator norm of T is equal to

sup{|λ(T (v))| : v ∈ V, λ ∈ W^{∗}, ∥v∥_{V} ≤ 1, ∥λ∥_{W^{∗}} ≤ 1}. (39.1)

Here ∥λ∥_{W^{∗}} denotes the dual norm of the bounded linear functional λ on W. It is clear from the definitions that (39.1) is less than or equal to the operator norm of T, and equality follows from the Hahn–Banach theorem.

The dual mapping T^{∗} sends a bounded linear functional λ on W to the bounded linear functional on V defined by

T^{∗}(λ) = λ ∘ T. (39.2)

It is easy to see that T^{∗} : W^{∗} → V^{∗} is a bounded linear operator with respect to the dual norms on W^{∗}, V^{∗}, whose operator norm is less than or equal to the operator norm of T. The operator norms of T and T^{∗} are actually the same, because of the characterization (39.1) of the operator norm of T.

For example, if V = W and T is the identity mapping on V, then T^{∗} is the identity mapping on V^{∗}. In the previous section, we saw that inclusion and quotient mappings are dual to each other.

Let V_{1}, V_{2}, and V_{3} be vector spaces, all real or all complex complex, and equipped with norms. Also let T_{1} : V_{1} → V_{2} and T_{2} : V_{2} → V_{3} be bounded linear mappings, so that T_{2} ∘ T_{1} is a bounded linear mapping from V_{1} into V_{3}. It follows easily from the definitions that

(T_{2} ∘ T_{1})^{∗} = T_{1}^{∗} ∘ T_{2}^{∗} (39.3)

as bounded linear mappings from V_{3}^{∗} into V_{1}^{∗}.

40 Second duals

Let V be a real or complex vector space with a norm ∥ · ∥, let V^{∗} be its dual space with the dual norm ∥ · ∥_{∗}, and let V^{∗∗} be the dual of V^{∗} with its dual norm ∥ · ∥_{∗∗}. As before, each v ∈ V determines a bounded linear functional η_{v} on V^{∗}, defined by η_{v}(λ) = λ(v) (40.1) for each λ ∈ V^{∗}, and the linear mapping v ↦ η_{v} is an isometric embedding from V into V^{∗∗}. Note that the weak topology on V corresponds exactly to the topology induced on the copy of V in V^{∗∗} by the weak^{∗} topology on V^{∗∗} as the dual of V^{∗}. A Banach space V is said to be reflexive if this embedding maps V onto V^{∗∗}, and finite-dimensional spaces are automatically reflexive.

Let λ_{1}, …, λ_{n} be finitely many bounded linear functionals on V, and consider W = {v ∈ V : λ_{1}(v) = ⋯ = λ_{n}(v) = 0}. (40.2)

This is a closed linear subspace of V of codimension ≤ n, which is to say that

dim V/W ≤ n. (40.3)

The space W^{⊥} of bounded linear functionals μ on V that vanish on W is spanned by λ_{1}, …, λ_{n}, and can be identified with (V/W)^{∗}, as in Section 38.

Let ϕ be a bounded linear functional on V^{∗}, and let ψ be its restriction to W^{⊥}. Since V/W is reflexive, ψ can be expressed by evaluation at an element of V/W whose quotient norm is equal to the norm of ψ as a linear functional on W^{⊥}, which is less than or equal to ∥ϕ∥_{∗∗}. It follows from the definition of the quotient norm that for each ϵ > 0 there is a v ∈ V such that

∥v∥ < ∥ϕ∥_{∗∗} + ϵ (40.4)

and ϕ(μ) = μ(v) for each μ ∈ W^{⊥}. In particular, the copy of V in V^{∗∗} is dense in V^{∗∗} with respect to the weak^{∗} topology on V^{∗∗} as the dual of V^{∗}, with control on the norms involved in the approximation of elements of V^{∗∗} by elements of the copy of V.

For example, let E be an infinite set, and let V be c_{0}(E). Thus V^{∗} can be identified with ℓ^{1}(E), and V^{∗∗} can be identified with ℓ^{∞}(E). The natural embedding of V into V^{∗∗} corresponds to the obvious inclusion of c_{0}(E) in ℓ^{∞}(E). In this case, the approximation of elements of ℓ^{∞}(E) by elements of c_{0}(E) can be seen quite concretely. It is a bit more convenient to think in terms of approximating f ∈ ℓ^{∞}(E) by functions of the form f_{A}, where A ⊆ E has only finitely many elements, f_{A} = f on A, and f_{A} = 0 on E\A. These approximations satisfy ∥f_{A}∥_{∞} ≤ ∥f∥_{∞} (40.5)

automatically. If h_{1}, …, h_{n} are summable functions on E, then

|∑_{x∈E} f(x) h_{j}(x) − ∑_{x∈E} f_{A}(x) h(x)| (40.6)

can be made arbitrarily small for j = 1, …, n, by making

∥f∥_{∞} ∑_{x∈E\A} |h_{j}(x)| (40.7)

arbitrarily small.

41 Continuous functions

Let X be a locally compact Hausdorff topological space, and let

C(X, R), C(X, C) (41.1)

be the vector spaces of continuous real or complex-valued functions on X. If K is a nonempty compact set in X, then

∥f∥_{K} = sup_{x∈K} |f(x)| (41.2)

defines a seminorm on these vector spaces. The collection of these seminorms satisfies the positivity condition that for each nonzero continuous function f on X there is a nonempty compact set K ⊆ X such that ∥f∥_{K} ≠ 0, and thus determines a nice topology on these vector spaces, as in Section 24. A sequence {f_{j}}_{j=1}^{∞} of continuous functions on X converges to another continuous function f on X in this topology if and only if {f_{j}}_{j=1}^{∞} converges to f uniformly on compact subsets of X.

If λ is a bounded linear functional on one of these vector spaces with respect to this collection of seminorms, then there is a nonempty compact set K ⊆ X and an A ≥ 0 such that |λ(f)| ≤ A ∥f∥_{K} (41.3)

for every continuous function f on X. This uses the fact that the union of finitely many compact subsets of X is a compact set to reduce finitely many seminorms to a single seminorm in this situation. In particular,

λ(f) = 0 (41.4)

when f = 0 on K.

Suppose that X is also σ-compact, so that there is a sequence of nonempty compact subsets K_{1}, K_{2}, … of X such that

X = ⋃_{j=1}^{∞} K_{j}. (41.5)

We may as well ask in addition that

K_{l} ⊆ K_{l+1} (41.6)

for each l ≥ 1, since otherwise we can replace K_{l} with

⋃_{j=1}^{l} K_{j} (41.7)

for each l. Using the fact that X is locally compact, one can improve this to get that K_{l} ⊆ K_{l+1}^{∘} (41.8) for each l, where E^{∘} denotes the interior of a set E ⊆ X. More precisely, local compactness implies that any compact set in X is contained in the interior of another compact set, and this can be applied repeatedly to get the previous condition. In particular,

X = ⋃_{j=1}^{∞} K_{j}^{∘}. (41.9)

If K ⊆ X is compact, then it follows that

K ⊆ K_{j}^{∘} ⊆ K_{j} (41.10)

for some j. Indeed, K is contained in the union of finitely many K_{j}^{∘}’s, by compactness, and therefore in a single K_{j}^{∘}, by monotonicity. In this case,

∥f∥_{K} ≤ ∥f∥_{K_{j}} (41.11)

for each continuous function f on X. This shows that the sequence of seminorms corresponding to this sequence of compact sets suffices to determine the same topology on the space of continuous functions on X. For example, if X = R^{n}, then one can take K_{j} to be the closed ball in R^{n} with respect to the standard metric centered at the origin and with radius j. Of course, if X is compact, then it suffices to take K = X, for which the corresponding seminorm is the supremum norm.

If X is equipped with the discrete topology, then every function on X is continuous, and only the finite subsets of X are compact. In this case, X is σcompact if and only if X has only finitely or countably many elements. For any X, it is easy to see that the continuous functions on X with compact support are dense among all continuous functions on X with respect to the topology described above. Indeed, if f is any continuous function on X and K ⊆ X is compact, then there is a continuous function g on X with compact support such that f = g on K. This follows from the fact that there is a continuous function h on X with compact support such that h = 1 on K, so that g = f h has compact support and is equal to f on K.

42 Bounded sets

Let W be a real or complex vector space. A set E ⊆ W is said to be bounded with respect to a seminorm N on W if N(w) is bounded for w ∈ E. Similarly, E is bounded with respect to a collection N of seminorms on W if E is bounded with respect to each N ∈ N.

Let V be a real or complex vector space equipped with a norm ∥v∥, let V^{∗} be the dual space of bounded linear functionals on V with the dual norm ∥λ∥_{∗}, and consider the collection of seminorms on V^{∗} given by

N_{v}^{∗}(λ) = |λ(v)|, v ∈ V. (42.1)

Any bounded set in V^{∗} with respect to the dual norm is automatically bounded with respect to this collection of seminorms. Conversely, if V is a Banach space, then any bounded set in V^{∗} with respect to this collection of seminorms is a bounded set with respect to the dual norm, as in Section 30. In the same way, a set in V is bounded with respect to the norm if and only if it is bounded with respect to the seminorms

N_{λ}(v) = |λ(v)|, λ ∈ V^{∗}. (42.2)

Let X be a locally compact Hausdorff topological space, and consider the vector space W of continuous real or complex-valued functions on X, as in the previous section. A set E ⊆ W is bounded with respect to the supremum seminorms over compact subsets of X if and only if for each nonempty compact set K ⊆ X there is a nonnegative real number C(K) such that

(42.3) |f(x)| ≤ C(K) for every f ∈ E and x ∈ K.

This does not mean that the functions f ∈ E have to be uniformly bounded on X, since C(K) depends on K.

The convex hull Ê of a set E in a real or complex vector space W consists of all finite sums of the form

∑_{j=1}^{n} t_{j} w_{j}, (42.4)

where t_{1}, …, t_{n} are nonnegative real numbers such that ∑_{j=1}^{n} t_{j} = 1, and w_{1}, …, w_{n} ∈ E. Thus Ê is a convex set, E ⊆ Ê, and Ê is contained in any convex set that contains E. If E is bounded with respect to a seminorm N on W, then it is easy to see that Ê is also bounded with respect to N. Hence Ê is bounded with respect to a collection N of seminorms on W when E is bounded with respect to N.

43 Hahn–Banach, revisited

Let V be a real or complex vector space, let W be a linear subspace of V, and let λ be a linear functional on W. Suppose that there is a nonnegative real number L and a seminorm N on V such that

|λ(w)| ≤ L N(w) (43.1)

for every w ∈ W. Under these conditions, there is an extension of λ to a linear functional μ on V such that

|μ(v)| ≤ L N(v) (43.2)

for every v ∈ V. This is the same as the earlier formulation of the Hahn-Banach theorem, except that N is allowed to be a seminorm instead of a norm. It is easy to see that the same arguments work in this case. Alternatively, if

Z = {z ∈ V : N(z) = 0}, (43.3) then Z is a linear subspace of V. The intersection of W and Z is contained in the kernel of λ, by (43.1), and one can first extend λ to the linear span of W and Z by setting equal to 0 on Z. This permits the problem to be transferred to the quotient space V/Z, on which N becomes a norm in an obvious way.

Suppose that N is a collection of seminorms on V that satisfies the usual positivity condition, so that for each v ∈ V with v ≠ 0 there is an N ∈ N such that N(v) > 0. Let V^{∗} be the vector space of continuous linear functionals on V with respect to the topology determined by N. If N consists of a single norm on V, then this is the same as the dual space defined in Section 14.

Let u be a nonzero vector in V, and fix N ∈ N such that N(u) > 0. If W is the 1-dimensional linear subspace of V spanned by u, and λ is the linear functional on W that satisfies λ(u) = N(u), then (43.1) holds with L = 1. Hence there is an extension of λ to a linear functional μ on V such that (43.2) holds with L = 1. In particular, μ ∈ V^{∗} and μ(u) ≠ 0.

Now let W_{0} be a closed linear subspace of V, and let u be a vector in V not in W_{0}. By the definition of the topology on V determined by N, there are finitely many seminorms N_{1}, …, N_{l} ∈ N and a positive real number r such that

max(N_{1}(u − w), …, N_{l}(u − w)) ≥ r (43.4)

for every w ∈ W_{0}. Put

N(v) = max(N_{1}(v), …, N_{l}(v)), (43.5)

which is also a seminorm on V. Let W be the linear subspace of V spanned by W_{0} and u, and let λ be the linear functional on W defined by

λ(t u + w) = t r (43.6)

for t ∈ R or C, as appropriate, and w ∈ W_{0}. Thus (43.4) implies that

|λ(t u + w)| ≤ N(t u + w) (43.7)

for every t ∈ R or C, as appropriate, and w ∈ W_{0}. This is the same as saying that (43.1) holds with L = 1. It follows that there is an extension of λ to a linear functional μ on V that satisfies (43.2) with L = 1.

References

[1] F. Albiac and N. Kalton, Topics in Banach Space Theory, Springer-Verlag, 2006.

[2] G. Bachman and L. Narici, Functional Analysis, Dover, 2000.

[3] R. Beals, Adavanced Mathematical Analysis, Springer-Verlag, 1973.

[4] R. Beals, Analysis: An Introduction, Cambridge University Press, 2004.

[5] B. Beauzamy, Introduction to Banach Spaces and their Geometry, 2nd edition, North-Holland, 1985.

[6] S. Bochner and K. Chandrasekharan, Fourier Transforms, Princeton University Press, 1949.

[7] A. Browder, Mathematical Analysis: An Introduction, Springer-Verlag, 1996.

[8] Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, American Mathematical Society, 2000.

[9] N. Carothers, A Short Course on Banach Space Theory, Cambridge University Press, 2005.

[10] J. Conway, A Course in Functional Analysis, 2nd edition, Springer-Verlag, 1990.

[11] C. Costara and D. Popa, Exercises in Functional Analysis, Kluwer, 2003.

[12] J. Diestel, Sequences and Series in Banach Spaces, Springer-Verlag, 1984.

[13] J. Dieudonné, History of Functional Analysis, North-Holland, 1981.

[14] R. Edwards, Functional Analysis: Theory and Applications, Dover, 1995.

[15] Y. Eidelman, V. Milman, and A. Tsolomitis, Functional Analysis: An Introduction, American Mathematical Society, 2004.

[16] G. Folland, Real Analysis: Modern Techniques and their Applications, 2nd edition, Wiley, 1999.

[17] T. Gamelin and R. Greene, Introduction to Topology, 2nd edition, Dover, 1999.

[18] R. Goldberg, Methods of Real Analysis, 2nd edition, Wiley, 1976.

[19] D. Ha, Functional Analysis: A Gentle Introduction, Matrix Editions, 2006.

[20] P. Halmos, Measure Theory, van Nostrand, 1950.

[21] V. Hansen, Functional Analysis: Entering Hilbert Space, World Scientific, 2006.

[22] A. Helemskii, Lectures and Exercises on Functional Analysis, translated from the 2004 Russian original by S. Akbarov, American Mathematical Society, 2006.

[23] E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer-Verlag, 1975.

[24] J. Horváth, Topological Vector Spaces and Distributions, Addison-Wesley, 1966.

[25] F. Jones, Lebesgue Integration on Euclidean Spaces, Jones and Bartlett, 1993.

[26] N. Kalton, N. Peck, and J. Roberts, An F -Space Sampler, Cambridge University Press, 1984.

[27] I. Kaplansky, Set Theory and Metric Spaces, 2nd edition, Chelsea, 1977.

[28] J. Kelley, General Topology, Springer-Verlag, 1975.

[29] J. Kelley, I. Namioka, et al., Linear Topological Spaces, Springer-Verlag, 1976.

[30] J. Kelley and T. Srinivasan, Measure and Integral, Springer-Verlag, 1988.

[31] A. Knapp, Basic Real Analysis, Birkhäuser, 2005.

[32] A. Knapp, Advanced Real Analysis, Birkhäuser, 2005.

[33] G. Köthe, Topological Vector Spaces I, II, Springer-Verlag, 1969, 1979.

[34] S. Krantz, A Panorama of Harmonic Analysis, Mathematical Association of America, 1999.

[35] S. Krantz, Real Analysis and Foundations, 2nd edition, Chapman & Hall / CRC, 2005.

[36] S. Krantz and H. Parks, The Geometry of Domains in Space, Birkhäuser, 1999.

[37] V. Krishnan, Textbook of Functional Analysis: A Problem-Oriented Approach, Prentice-Hall of India, 2001.

[38] B. Lahiri, Elements of Functional Analysis, 6th edition, World Press, 2005.

[39] S. Lang, Real and Functional Analysis, 3rd edition, Springer-Verlag, 1993.

[40] S. Lang, Undergraduate Analysis, 2nd edition, Springer-Verlag, 1997.

[41] P. Lax, Functional Analysis, Wiley, 2002.

[42] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, Lecture Notes in Mathematics 338, Springer-Verlag, 1973.

[43] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I: Sequence Spaces, Springer-Verlag, 1977.

[44] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II: Function Spaces, Springer-Verlag, 1979.

[45] R. Megginson, An Introduction to Banach Space Theory, Springer-Verlag, 1998.

[46] P. Meyer-Nieberg, Banach Lattices, Springer-Verlag, 1991.

[47] V. Milman and G. Schechtman, Asymptotic Theory of Finite-Dimensional Normed Spaces, Lecture Notes in Mathematics 1200, 1986.

[48] A. Monna, Functional Analysis in Historical Perspective, Wiley, 1973.

[49] T. Morrison, Functional Analysis: An Introduction to Banach Space Theory, Wiley, 2001.

[50] A. Pietsch, History of Banach Spaces and Linear Operators, Birkhäuser, 2007.

[51] A. Pietsch and J. Wenzel, Orthonormal Systems and Banach Space Geometry, Cambridge University Press, 1998.

[52] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge University Press, 1989.

[53] S. Ponnusamy, Foundations of Functional Analysis, Alpha Science International, 2002.

[54] S. Promislow, A First Course in Functional Analysis, Wiley, 2008.

[55] M. Reed, Fundamental Ideas of Analysis, Wiley, 1998.

[56] F. Riesz and B. Sz.-Nagy, Functional Analysis, translated from the second French edition by L. Boron, Dover, 1990.

[57] A. Robertson and W. Robertson, Topological Vector Spaces, 2nd edition, Cambridge University Press, 1980.

[58] H. Royden, Real Analysis, 3rd edition, Macmillan, 1988.

[59] W. Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976.

[60] W. Rudin, Real and Complex Analysis, 3rd edition, McGraw-Hill, 1987.

[61] W. Rudin, Functional Analysis, 2nd edition, McGraw-Hill, 1991.

[62] B. Rynne and M. Youngson, Linear Functional Analysis, 2nd edition, Springer-Verlag, 2008.

[63] K. Saxe, Beginning Functional Analysis, Springer-Verlag, 2002.

[64] H. Schaefer and M. Wolff, Topological Vector Spaces, 2nd edition, Springer-Verlag, 1999.

[65] M. Schechter, Principles of Functional Analysis, 2nd edition, American Mathematical Society, 2002.

[66] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.

[67] E. Stein and R. Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, 2005.

[68] E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971.

[69] R. Strichartz, The Way of Analysis, Jones and Bartlett, 1995.

[70] K. Stromberg, Introduction to Classical Real Analysis, Wadsworth, 1981.

[71] D. Stroock, A Concise Introduction to the Theory of Integration, 3rd edition, Birkhäuser, 1999.

[72] E. Suhubi, Functional Analysis, Kluwer, 2003.

[73] M. Taylor, Measure Theory and Integration, American Mathematical Society, 2006.

[74] A. Thompson, Minkowski Geometry, Cambridge University Press, 1996.

[75] F. Trèves, Topological Vector Spaces, Distributions, and Kernels, Dover, 2006.

[76] R. Wheeden and A. Zygmund, Measure and Integral: An Introduction to Real Analysis, Dekker, 1977.

[77] A. Wilansky, Modern Methods in Topological Vector Spaces, McGraw-Hill, 1978.

[78] P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge University Press, 1991.

[79] K. Yosida, Functional Analysis, Springer-Verlag, 1995.