Completeness

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.