Frechet and countably normed spaces

Suppose in a countably-normed space $X$ each subsequent seminorm majorizes the preceding one, but is not equivalent to it. Prove that $X$ is not normable.

Prove that the spaces $s, l_{2}^{\infty }, C^{\infty }[0,1], \mathcal{A}(D)$ and $C(\mathbb {R})$ are separable Fréchet spaces.

Let $X$ be a countably-normed space, and let $Y$ be a proper closed subspace of it. A vector $x \in X$ is called an $\varepsilon$-perpendicular to $Y$ if $d(x, 0) = 1$ and $d(x, Y) > 1-\varepsilon$. Prove that an $\varepsilon$-perpendicular to $Y$ exists for any $\varepsilon > 0$.

Let $X$ be a countably-normed Fréchet space. Prove that any system of nested closed balls of the space $X$ has nonempty intersection (compare with problem 2.68).

Let $X$ be a countably-normed Fréchet space. Prove that if $Y$ is a finite-dimensional subspace of $X$, and $x$ is an arbitrary vector from $X$, then there is a vector $y_{0}$ in $Y$ for which $\operatorname {dist}(x, Y) = d\left(x, y_{0}\right)$ (compare with problem 2.70).

(Analogue of the Banach theorem on the inverse operator). Let $X$ and $Y$ be countably-normed Fréchet spaces, and $T: X \rightarrow Y$ a linear continuous bijective operator. Prove that $T$ has a linear continuous inverse operator.

(Analogue of the Banach-Steinhaus theorem). Let $\mathcal{M}$ be an arbitrary family of continuous linear operators from a countably-normed Fréchet space $(X,\left\{ p_{n}\right\}_{1}^{\infty })$ into a countably-normed Fréchet space $(Y,\left\{ q_{n}\right\}_{1}^{\infty })$, such that for every $x \in X$ and $n \in \mathbb {N}$ the set $\left\{ q_{n}(T x): T \in \mathcal{M}\right\}$ is bounded. Prove that for every seminorm $q_{n}$ there exist seminorms $p_{k_{1}}, \ldots , p_{k_{j}}$ and a constant $C_{n} > 0$ such that $q_{n}(T x) \leqslant C_{n} \max \left\{ p_{k_{1}}(x), \ldots , p_{k_{j}}(x)\right\}$ for all $x \in X$ and $T \in \mathcal{M}$.

Prove that the spaces $\mathcal{E}$ and $\mathcal{S}$ are metrizable (and, moreover, are Fréchet spaces), while the space $\mathcal{D}$ is not metrizable.

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.

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.

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?

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

Prove the Open Mapping Theorem for Fréchet spaces.

Show that the space $s$ of all sequences of scalars (with the metric $d(x,y) = \sum_{n=1}^{\infty } 2^{-n} \left|x(n)-y(n)\right| / (1 + \left|x(n)-y(n)\right|)$) is linearly homeomorphic to $C(\mathbb {N})$ and describe $s^*$.

In the space $C(\mathbb {R})$ of all continuous real functions on the line introduce the countable system of seminorms $p_N(f)=\max_{\left|x\right|\leq N}\left|f(x)\right|$ and set

For what $r>0$ is the ball of radius $r$ a convex set in $C(\mathbb {R})$?

Let $L$ be a complete LCS and $\Omega$ a domain in $\mathbb {R}^{n}$. We define the space $\mathcal E(\Omega ,L)$ as the collection of infinitely differentiable functions on $\Omega$ with values in $L$. If $\left\{ p_\alpha \right\}_{\alpha \in A}$ is the family of seminorms that determines the topology in $L$, then the topology in $\mathcal E(\Omega ,L)$ is given by the family of seminorms

where $K$ is a compact subset of $\Omega$ and $l=(l_1,\ldots ,l_n)$ is a multi-index, with $\alpha \in A$. Prove that $\mathcal E(\Omega ,L)$ is a complete LCS, metrizable if $L$ is metrizable.

Let $\Omega_1$ and $\Omega_2$ be domains in $\mathbb {R}^{n}$ and $\mathbb {R}^{m}$. Then $\Omega_1\times \Omega_2$ is a domain in $\mathbb {R}^{n+m}$. In the notation of Problem 427 prove that

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 $\alpha$.

Suppose $\mu$ is a Borel probability measure on a compact Hausdorff space $Q, X$ is a Fréchet space, and $f: Q \rightarrow X$ is continuous. A partition of $Q$ is, by definition, a finite collection of disjoint Borel subsets of $Q$ whose union is $Q$. Prove that to every neighborhood $V$ of 0 in $X$ there corresponds a partition $\left\{ E_{i}\right\}$ such that the difference

lies in $V$ for every choice of $s_{i} \in E_{i}$. (This exhibits the integral as a strong limit of "Riemann sums".) Suggestion: Take $V$ convex and balanced. If $\Lambda \in X^{*}$ and if $\left|\Lambda x\right| \leq 1$ for every $x \in V$, then $\left|\Lambda z\right| \leq 1$, provided that the sets $E_{i}$ are chosen so that $f(s)-f(t) \in V$ whenever $s$ and $t$ lie in the same $E_{i}$.

Suppose $\Omega$ is a region in $\mathbb {C}, X$ is a Fréchet space, and $f: \Omega \rightarrow X$ is holomorphic.