Math Problems Vault

Problem IV.2.1

Let $\left\{ p_1, p_2, \dots \right\}$ be a sequence of seminorms on $\mathscr {X}$ such that $\bigcap_{n=1}^{\infty } \left\{ x : p_n(x) = 0\right\} = (0)$ . For $x$ and $y$ in $\mathscr {X}$ , define

d(x,y) = \sum _{n=1}^{\infty } 2^{-n} \frac{p_n(x-y)}{1 + p_n(x-y)}.

Then $d$ is a metric on $\mathscr {X}$ and the topology on $\mathscr {X}$ defined by $d$ is the topology on $\mathscr {X}$ defined by the seminorms $\left\{ p_1, p_2, \dots \right\}$ . Thus a locally convex space is metrizable if and only if its topology is determined by a countable family of seminorms. Supply the missing details in the proof of this proposition: show that $d$ is a metric and induces the same topology as $\left\{ p_n\right\}$ .

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Problem IV.2.2

Let $X$ be completely regular and let $C(X)$ be all continuous functions from $X$ into $\mathbb {F}$ , topologized by the seminorms $p_K(f) = \sup \left\{ \left|f(x)\right| : x \in K\right\}$ for $K$ compact in $X$ . Then $C(X)$ is metrizable if and only if $X = \bigcup_{n=1}^{\infty } K_n$ , where each $K_n$ is compact, $K_1 \subseteq K_2 \subseteq \cdots$ , and if $K$ is any compact subset of $X$ , then $K \subseteq K_n$ for some $n$ . Verify this statement.

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Problem IV.2.3

Let $X$ be locally compact and let $C(X)$ be as in Exercise 2 above. Then $C(X)$ is metrizable if and only if $X$ is $\sigma$ -compact (that is, $X$ is the union of a sequence of compact sets). Also, the space $H(G)$ of analytic functions on an open set $G \subseteq \mathbb {C}$ , topologized by uniform convergence on compact subsets, is metrizable. Verify these statements.

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Problem IV.2.4

Let $\mathscr {X}$ be a TVS and prove the following.

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(a)

If $B$ is a bounded subset of $\mathscr {X}$ , then so is $\operatorname {cl} B$ .

(b)

The finite union of bounded sets is bounded.

(c)

Every compact set is bounded.

(d)

If $B \subseteq \mathscr {X}$ , then $B$ is bounded if and only if for every sequence $\left\{ x_n\right\}$ contained in $B$ and for every $\left\{ \alpha_n\right\}$ in $c_0$ , $\alpha_n x_n \to 0$ in $\mathscr {X}$ .

(e)

If $\mathscr {Y}$ is a TVS, $T : \mathscr {X} \to \mathscr {Y}$ is a continuous linear transformation, and $B$ is a bounded subset of $\mathscr {X}$ , then $T(B)$ is a bounded subset of $\mathscr {Y}$ .

(f)

If $\mathscr {X}$ is a LCS and $B \subseteq \mathscr {X}$ , then $B$ is bounded if and only if for every continuous seminorm $p$ , $\sup \left\{ p(b) : b \in B\right\} < \infty$ .

(g)

If $\mathscr {X}$ is a normed space and $B \subseteq \mathscr {X}$ , then $B$ is bounded if and only if $\sup \left\{ \left\| b\right\| : b \in B\right\} < \infty$ .

(h)

If $\mathscr {X}$ is a Fréchet space, then bounded sets have finite diameter, but not conversely.

(i)

The translate of a bounded set is bounded.

Problem IV.2.5

If $\mathscr {X}$ is a LCS, show that $\mathscr {X}$ is metrizable if and only if $\mathscr {X}$ is first countable. Is this equivalent to saying that $\left\{ 0\right\}$ is a $G_\delta$ set?

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Problem IV.2.6

Let $X$ be a locally compact space and give $C_b(X)$ the strict topology $\beta$ : for each $\phi$ in $C_0(X)$ , define the seminorm $p_\phi (f) = \left\| \phi f\right\|_\infty$ for $f$ in $C_b(X)$ , and let $\beta$ be the topology defined by these seminorms. Show that a subset of $C_b(X)$ is $\beta$ -bounded if and only if it is norm bounded.

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Problem IV.2.7

With the notation of Exercise 6, show that $(C_b(X), \beta )$ is metrizable if and only if $X$ is compact.

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Problem IV.2.8

Prove the Open Mapping Theorem for Fréchet spaces.

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