General Theory

Suppose $X$ is a vector space. All sets mentioned below are understood to be subsets of $X$. Prove the following statements from the axioms as given in Section 1.4. (Some of these are tacitly used in the text).

The convex hull of a set $A$ in a vector space $X$ is the set of all convex combinations of members of $A$, that is, the set of all sums

in which $x_{i} \in A, t_{i} \geq 0, \sum t_{i} = 1 ; n$ is arbitrary. Prove that the convex hull of $A$ is convex and that it is the intersection of all convex sets that contain $A$.

Let $X$ be a topological vector space. All sets mentioned below are understood to be the subsets of $X$. Prove the following statements.

Let $B = \left\{ \left(z_{1}, z_{2}\right) \in \mathbb {C}^{2}:\left|z_{1}\right| \leq \left|z_{2}\right|\right\}$. Show that $B$ is balanced but that its interior is not.

Consider the definition of "bounded set" given in Section 1.6. Would the content of this definition be altered if it were required merely that to every neighborhood $V$ of 0 corresponds some $t > 0$ such that $E \subset t V$?

Prove that a set $E$ in a topological vector space is bounded if and only if every countable subset of $E$ is bounded.

Let $X$ be the vector space of all complex functions on the unit interval $[0,1]$, topologized by the family of seminorms

This topology is called the topology of pointwise convergence. Justify this terminology.

Show that there is a sequence $\left\{ f_{n}\right\}$ in $X$ such that

Suppose

Suppose $X$ and $Y$ are topological vector spaces, $\operatorname {dim} Y < \infty , \Lambda : X \rightarrow Y$ is linear, and $\Lambda (X) = Y$.

If $N$ is a subspace of a vector space $X$, the codimension of $N$ in $X$ is, by definition, the dimension of the quotient space $X / N$.

Suppose $0 < p < 1$ and prove that every subspace of finite codimension is dense in $L^{p}$. (See Section 1.47).

Suppose $d_{1}(x, y) = \left|x-y\right|, d_{2}(x, y) = \left|\phi (x)-\phi (y)\right|$, where $\phi (x) = x /(1+\left|x\right|)$. Prove that $d_{1}$ and $d_{2}$ are metrics on $\mathbb {R}$ which induce the same topology, although $d_{1}$ is complete and $d_{2}$ is not.

Let $C$ be the vector space of all complex continuous functions on [0,1]. Define

Let $(C, \sigma )$ be $C$ with the topology induced by this metric. Let $(C, \tau )$ be the topological vector space defined by the seminorms

in accordance with Theorem 1.37.

Put $K = [0,1]$ and define $\mathscr {D}_{K}$ as in Section 1.46. Show that the following three families of seminorms (where $n = 0,1,2, \ldots$) define the same topology on $\mathscr {D}_{K}$, if $D = d / d x$ :

Prove that the spaces $C(\Omega )$ (Section 1.44) do not have the Heine-Borel property.

Prove that the topology of $C(\Omega )$ does not depend on the particular choice of $\left\{ K_{n}\right\}$, as long as this sequence satisfies the conditions specified in Section 1.44. Do the same for $C^{\infty }(\Omega )$ (Section 1.46).

In the setting of Section 1.46, prove that $f \rightarrow D^{\alpha } f$ is a continuous mapping of $C^{\infty }(\Omega )$ into $C^{\infty }(\Omega )$ and also of $\mathscr {D}_{K}$ into $\mathscr {D}_{K}$, for every multi-index $\alpha$.

Prove the proposition concerning addition in the binary system which was used at the end of the proof of Theorem 1.24.

Suppose $M$ is a dense subspace of a topological vector space $X, Y$ is an $F$-space, and $\Lambda : M \rightarrow Y$ is continuous (relative to the topology that $M$ inherits from $X$) and linear. Prove that $\Lambda$ has a continuous linear extension $\overline{\Lambda }: X \rightarrow Y$.

Suggestion: Let $V_{n}$ be balanced neighborhoods of 0 in $X$ such that $V_{n}+V_{n} \subset V_{n-1}$ and such that $d(0, \Lambda x) < 2^{-n}$ if $x \in M \cap V_{n}$. If $x \in X$ and $x_{n} \in \left(x+V_{n}\right) \cap M$, show that $\left\{ \Lambda x_{n}\right\}$ is a Cauchy sequence in $Y$, and define $\overline{\Lambda } x$ to be its limit. Show that $\overline{\Lambda }$ is well defined, that $\overline{\Lambda } x = \Lambda x$ if $x \in M$, and that $\overline{\Lambda }$ is linear and continuous.

For each real number $t$ and each integer $n$, define $e_{n}(t) = e^{i n t}$, and define

Regard these functions as members of $L^{2}(-\pi , \pi )$. Let $X_{1}$ be the smallest closed subspace of $L^{2}$ that contains $e_{0}, e_{1}, e_{2}, \ldots$, and let $X_{2}$ be the smallest closed subspace of $L^{2}$ that contains $f_{1}, f_{2}, f_{3}, \ldots$ Show that $X_{1}+X_{2}$ is dense in $L^{2}$ but not closed. For instance, the vector

is in $L^{2}$ but not in $X_{1}+X_{2}$. (Compare with Theorem 1.42).

Let $V$ be a neighborhood of 0 in a topological vector space $X$. Prove that there is a real continuous function $f$ on $X$ such that $f(0) = 0$ and $f(x) = 1$ outside $V$. (Thus $X$ is a completely regular topological space). Suggestion: Let $V_{n}$ be balanced neighborhoods of 0 such that $V_{1}+V_{1} \subset V$ and $V_{n+1}+V_{n+1} \subsetneq V_{n}$. Construct $f$ as in the proof of Theorem 1.24. Show that $f$ is continuous at 0 and that

If $f$ is a complex function defined on the compact interval $I = [0,1] \subset R$, define

If $0 < \alpha \leq 1$, the corresponding Lipschitz space Lip $\alpha$ consists of all $f$ for which

is finite. Define

Prove that $\operatorname {Lip} \alpha$ is a Banach space and that $\operatorname {lip} \alpha$ is a closed subspace of $\operatorname {Lip} \alpha$.

Let $X$ be the vector space of all continuous functions on the open segment $(0,1)$. For $f \in X$ and $r > 0$, let $V(f, r)$ consist of all $g \in X$ such that $\left|g(x)-f(x)\right| < r$ for all $x \in (0,1)$. Let $\tau$ be the topology on $X$ that these sets $V(f, r)$ generate. Show that addition is $\tau$-continuous but scalar multiplication is not.

Show that the set $W$ that occurs in the proof of Theorem 1.14 need not be convex, and that $A$ need not be balanced unless $U$ is convex.

If $X$ is an infinite-dimensional topological vector space which is the union of countably many finite-dimensional subspaces, prove that $X$ is of the first category in itself. Prove that therefore no infinite-dimensional $F$-space has a countable Hamel basis. (A set $\beta$ is a Hamel basis for a vector space $X$ if $\beta$ is a maximal linearly independent subset of $X$. Alternatively, $\beta$ is a Hamel basis if every $x \in X$ has a unique representation as a finite linear combination of elements of $\beta$).

Sets of first and second category are "small" and "large" in a topological sense. These notions are different when "small" and "large" are understood in the sense of measure, even when the measure is intimately related to the topology. To see this, construct a subset of the unit interval which is of the first category but whose Lebesgue measure is 1.

Put $K = [-1,1]$; define $\mathscr {D}_{K}$ as in Section 1.46 (with $\mathbb {R}$ in place of $\mathbb {R}^{n}$). Suppose $\left\{ f_{n}\right\}$ is a sequence of Lebesgue integrable functions such that

exists for every $\phi \in \mathscr {D}_{K}$. Show that $\Lambda$ is a continuous linear functional on $\mathscr {D}_{K}$. Show that there is a positive integer $p$ and a number $M < \infty$ such that

for all $n$. For example, if $f_{n}(t) = n^{3} t$ on $[-1 / n, 1 / n]$ and 0 elsewhere, show that this can be done with $p = 1$. Construct an example where it can be done with $p = 2$ but not with $p = 1$.

Let $L^{1}$ and $L^{2}$ be the usual Lebesgue spaces on the unit interval. Prove that $L^{2}$ is of the first category in $L^{1}$, in three ways:

Prove results analogous to those of Exercise 4 for the spaces $\ell^{p}$, where $\ell^{p}$ is the Banach space of all complex functions $x$ on $\left\{ 0,1,2, \ldots \right\}$ whose norm

is finite.

Define the Fourier coefficients $\hat{f}(n)$ of a function $f \in L^{2}(T)(T$ is the unit circle) by

for all $n \in Z$ (the integers). Put

Prove that $\left\{ f \in L^{2}(T): \lim_{n \rightarrow \infty } \Lambda_{n} f \text{ exists }\right\}$ is a dense subspace of $L^{2}(T)$ of the first category.

Let $C(T)$ be the set of all continuous complex functions on the unit circle $T$. Suppose $\left\{ \gamma_{n}\right\} (n \in Z)$ is a complex sequence that associates to each $f \in C(T)$ a function $\Lambda f \in C(T)$ whose Fourier coefficients are

(The notation is as in Exercise 6). Prove that $\left\{ \gamma_{n}\right\}$ has this multiplier property if and only if there is a complex Borel measure $\mu$ on $T$ such that

Suggestion: With the supremum norm, $C(T)$ is a Banach space. Apply the closed graph theorem. Then consider the functional

and apply the Riesz representation theorem ([23], Th. 6.19). (The above series may not converge; use it only for trigonometric polynomials).

Define functionals $\Lambda_{m}$ on $\ell^{2}$ (see Exercise 5) by

Define $x_{n} \in \ell^{2}$ by $x_{n}(n) = 1 / n, x_{n}(i) = 0$ if $i \neq n$. Let $K \subset \ell^{2}$ consist of $0, x_{1}, x_{2}$, $x_{3}, \ldots$ Prove that $K$ is compact. Compute $\Lambda_{m} x_{n}$. Show that $\left\{ \Lambda_{m} x\right\}$ is bounded for each $x \in K$ but $\left\{ \Lambda_{m} x_{m}\right\}$ is not. Convexity can therefore not be omitted from the hypotheses of Theorem 2.9.

Choose $c_{n} > 0$ so that $\sum c_{n} = 1, \sum n c_{n} = \infty$. Take $x = \sum c_{n} x_{n}$. Show that $x$ lies in the closed convex hull of $K$ (by definition, this is the closure of the convex hull) and that $\left\{ \Lambda_{m} x\right\}$ is not bounded.

Show that the convex hull of $K$ is not closed.

Suppose $X, Y, Z$ are Banach spaces and

is bilinear and continuous. Prove that there exists $M < \infty$ such that

Is completeness needed here?

Prove that a bilinear mapping is continuous if it is continuous at the origin $(0,0)$.

Define $B\left(x_{1}, x_{2} ; y\right) = \left(x_{1} y, x_{2} y\right)$. Show that $B$ is a bilinear continuous mapping of $\mathbb {R}^{2} \times R$ onto $\mathbb {R}^{2}$ which is not open at $(1,1 ; 0)$. Find all points where this $B$ is open.

Let $X$ be the normed space of all real polynomials in one variable, with

Put $B(f, g) = \int_{0}^{1} f(t) g(t) d t$, and show that $B$ is a bilinear functional on $X \times X$ which is separately continuous but is not continuous.

Suppose $X$ is a topological vector space which is of the second category in itself. Let $K$ be a closed, convex, absorbing subset of $X$. Prove that $K$ contains a neighborhood of 0. Suggestion: Show first that $H = K \cap (-K)$ is absorbing. By a category argument, $H$ has interior. Then use

Show that the result is false without convexity of $K$, even if $X = \mathbb {R}^{2}$. Show that the result is false if $X$ is $L^{2}$ topologized by the $L^{1}$-norm (as in Exercise 4).

Suppose $X$ is an $F$-space and $Y$ is a subspace of $X$ whose complement is of the first category. Prove that $Y = X$. Hint: $Y$ must intersect $x+Y$ for every $x \in X$.

Suppose that $X$ and $K$ are metric spaces, that $K$ is compact, and that the graph of $f: X \rightarrow K$ is a closed subset of $X \times K$. Prove that $f$ is continuous. (This is an analogue of Theorem 2.15 but is much easier). Show that compactness of $K$ cannot be omitted from the hypotheses, even when $X$ is compact.

Call a set $H \subset \mathbb {R}^{n}$ a hyperplane if there exist real numbers $a_{1}, \ldots , a_{n}, c$ (with $a_{i} \neq 0$ for at least one $i$) such that $H$ consists of all points $x = \left(x_{1}, \ldots , x_{n}\right)$ that satisfy $\sum a_{i} x_{i} = c$.

Suppose $E$ is a convex set in $\mathbb {R}^{n}$, with nonempty interior, and $y$ is a boundary point of $E$. Prove that there is a hyperplane $H$ such that $y \in H$ and $E$ lies entirely on one side of $H$. (State the conclusion more precisely). Suggestion: Suppose 0 is an interior point of $E$, let $M$ be the one-dimensional subspace that contains $y$, and apply Theorem 3.2.

Suppose $L^{2} = L^{2}([-1,1])$, with respect to Lebesgue measure. For each scalar $\alpha$, let $E_{\alpha }$ be the set of all continuous functions $f$ on $[-1,1]$ such that $f(0) = \alpha$. Show that each $E_{\alpha }$ is convex and that each is dense in $L^{2}$. Thus $E_{\alpha }$ and $E_{\beta }$ are disjoint convex sets (if $\alpha \neq \beta$) which cannot be separated by any continuous linear functional $\Lambda$ on $L^{2}$. Hint: What is $\Lambda \left(E_{\alpha }\right)$?

Suppose $X$ is a real vector space (without topology). Call a point $x_{0} \in A \subset X$ an internal point of $A$ if $A-x_{0}$ is an absorbing set.

Let $\ell^{\infty }$ be the space of all real bounded functions $x$ on the positive integers. Let $\tau$ be the translation operator defined on $\ell^{\infty }$ by the equation

Prove that there exists a linear functional $\Lambda$ on $\ell^{\infty }$ (called a Banach limit) such that

For $0 < p < \infty$, let $\ell^{p}$ be the space of all functions $x$ (real or complex, as the case may be) on the positive integers, such that

For $1 \leq p < \infty$, define $\left\| x\right\|_{p} = \left\{ \sum \left|x(n)\right|^{p}\right\}^{1 / p}$, and define $\left\| x\right\|_{\infty } = \sup_{n}\left|x(n)\right|$.

Put $f_{n}(t) = e^{i n t}(-\pi \leq t \leq \pi )$; let $L^{p} = L^{p}(-\pi , \pi )$, with respect to Lebesgue measure. If $1 \leq p < \infty$, prove that $f_{n} \rightarrow 0$ weakly in $L^{p}$, but not strongly.

$L^{\infty }([0,1])$ has its norm topology $\left(\left\| f\right\|_{\infty }\right.$ is the essential supremum of $\left.\left|f\right|\right)$ and its weak*-topology as the dual of $L^{1}$. Show that $C$, the space of all continuous functions on $[0,1]$, is dense in $L^{\infty }$ in one of these topologies but not in the other. (Compare with the corollaries to Theorem 3.12). Show the same with "closed " in place of "dense".

Let $C$ be the Banach space of all complex continuous functions on [0,1], with the supremum norm. Let $B$ be the closed unit ball of $C$. Show that there exist continuous linear functionals $\Lambda$ on $C$ for which $\Lambda (B)$ is an open subset of the complex plane; in particular, $\left|\Lambda \right|$ attains no maximum on $B$.

Let $E \subset L^{2}(-\pi , \pi )$ be the set of all functions

where $m, n$ are integers and $0 \leq m < n$. Let $E_{1}$ be the set of all $g \in L^{2}$ such that some sequence in $E$ converges weakly to $g$. ($E_{1}$ is called the weak sequential closure of E).

Represent $\ell^{1}$ as the space of all real functions $x$ on $S = \left\{ (m, n): m \geq 1, n \geq 1\right\}$, such that

Let $c_{0}$ be the space of all real functions $\gamma$ on $S$ such that $y(m, n) \rightarrow 0$ as $m+n \rightarrow \infty$, with norm $\left\| y\right\|_{\infty } = \sup \left|y(m, n)\right|$.

Let $M$ be the subspace of $\ell^{1}$ consisting of all $x \in \ell^{1}$ that satisfy the equations