Locally convex spaces as polynormed spaces

Prove that a polynormed space is locally convex.

Let $X$ be a linear topological locally convex space. Prove that the topology on $X$ can be defined by some system of seminorms, i.e., $X$ can be regarded as a polynormed space.

Prove that convergence in the spaces $s, \mathcal{A}(D), C^{\infty }[0,1]$, $C(\mathbb {R})$ and $C_{\mathrm{e}}(\mathbb {R})$ is given by the system of seminorms indicated in the list of spaces (note that for all spaces except $C_{\mathrm{e}}(\mathbb {R})$, this system of seminorms is countable).

Prove that the topology of a polynormed space $(X,\left\{ p_{\alpha }\right\}_{\alpha \in A})$ with the base $\mathcal{B}$ of neighborhoods of zero indicated in the preamble is the intersection $\bigcap \tau$ of all topologies $\tau$ with respect to which all the seminorms $p_{\alpha }, \alpha \in A$, are continuous.

Prove that a sequence $\left\{ x_{n}\right\}_{1}^{\infty }$ in a polynormed space $(X,\left\{ p_{\alpha }\right\}_{\alpha \in A})$ converges (in the sense of the topology generated by the system of seminorms) to a vector $x$ if and only if $p_{\alpha }\left(x_{n}-x\right) \rightarrow 0$ with respect to every seminorm $p_{\alpha }$.

Prove that a polynormed space $X$ is Hausdorff with respect to the topology generated by the system of its seminorms $\left\{ p_{\alpha }\right\}_{\alpha \in A}$ if and only if for every vector $x \in X, x \neq 0$, there exists a seminorm $p_{\alpha }, \alpha = \alpha (x)$, such that $p_{\alpha }(x) > 0$.

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

Let a linear space $X$ be given two families of seminorms $\left\{ p_{\alpha }\right\}_{\alpha \in A_{1}}$ and $\left\{ p_{\alpha }\right\}_{\alpha \in A_{2}}$. Denote the topologies generated by these families by $\tau_{1}$ and $\tau_{2}$. Prove that $\tau_{1}$ is weaker than $\tau_{2}$ if and only if the second family of seminorms majorizes the first.

Without using Theorem 14.2, prove that a Hausdorff polynormed space $X$ is normable if and only if its family of seminorms is equivalent to a finite subfamily of itself.

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 $s, l_{2}^{\infty }, C^{\infty }[0,1], \mathcal{A}(D), C(\mathbb {R})$ and $C_{\mathrm{e}}[0,1]$ are not normable.

Without using Theorem 14.1, prove that a Hausdorff polynormed space is metrizable if and only if its family of seminorms is equivalent to an at most countable subfamily of itself. In particular, any Hausdorff countably-normed space is metrizable.

Prove that the space $C_{\mathrm{e}}[0,1]$ is not metrizable.

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

Prove that a set $E$ in a polynormed space $(X,\left\{ p_{\alpha }\right\}_{\alpha \in A})$ is bounded in the sense of Definition 14.3 if and only if for every $\alpha \in A$ there exists a constant $C_{\alpha } > 0$ such that $\sup \left\{ p_{\alpha }(x): x \in E\right\} < C_{\alpha }$.

Definition 14.3, the relevant part: A subset $E$ of a linear topological space $X$ is called bounded if for every neighborhood of zero $U \subset X$ there is a number $s > 0$ such that $E \subset t U$ for every $t > s$.

Prove that a set $M$ in the space $s$ is precompact if and only if it is bounded, i.e., for every $k \in \mathbb {N}$ there exists $A_{k} > 0$ such that $\left|x_{k}\right| \leqslant A_{k}$ for all $x = \left(x_{1}, x_{2}, \ldots \right) \in M$.

Prove that a set $M$ in the space $C^{\infty }[0,1]$ is precompact if and only if it is bounded, i.e., for every $k = 0,1,2, \ldots$ there exists $A_{k} > 0$ such that $\max_{t \in [0,1]}\left|x^{(k)}(t)\right| \leqslant A_{k}$ for all $x \in M$.

(Compactness principle, P. Montel, 1910). Prove that a set $M$ in the space $\mathcal{A}(D)$ is precompact if and only if it is bounded in the space $\mathcal{A}(D)$ (i.e., $\sup \left\{ \left|f(z)\right|: f \in M, z \in Q\right\} < \infty$ for every compact set $Q \subset D$).

Are the sets:

Is the set of functions holomorphic on the segment $[0,1]$ a closed subspace of the space $C^{\infty }[0,1]$?

Prove that the linear subspace of continuous bounded functions on $\mathbb {R}$ is a non-closed, everywhere dense subspace of the polynormed space $C(\mathbb {R})$.

Prove that any finite-dimensional subspace of a Hausdorff polynormed space is closed.

Let $Y$ be a closed subspace of a polynormed space $(X,\left\{ p_{\alpha }\right\}_{\alpha \in A})$. Prove that

are seminorms on the linear space $X / Y$.

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).

Let $f$ be a linear functional on a polynormed space $(X,\left\{ p_{\alpha }\right\}_{\alpha \in A})$. Prove the equivalence of the following conditions:

1. $f$ is continuous at every point $x \in X$;

2. $f$ is continuous at zero;

3. there exist $\alpha_{1}, \ldots \alpha_{n} \in A$ and a constant $C > 0$ such that for every $x \in X$ the inequality

   $$\left|f(x)\right| \leqslant C \max \left\{ p_{\alpha _{1}}(x), \ldots , p_{\alpha _{n}}(x)\right\}$$

   holds.

Give an example of a polynormed space $(X,\left\{ p_{\alpha }\right\}_{\alpha \in A})$ and a continuous linear functional $f$ on it such that the estimate $\left|f(x)\right| \leqslant C p_{\alpha }(x)$ does not hold for any seminorm $p_{\alpha }$.

(Analogue of the Hahn-Banach theorem). Let $(X,\left\{ p_{\alpha }\right\}_{\alpha \in A})$ be a Hausdorff polynormed space. Prove that if $f$ is a linear continuous complex-valued functional, defined on a linear subspace $Y$ of the space $X$, then there exists a linear continuous functional $F$, defined on all of $X$ and coinciding with $f$ on the subspace $Y$. Deduce from this that for every $x \in X, x \neq 0$, there exists a linear continuous functional $f$ on $X$ such that $f(x) \neq 0$.

Prove that the functional $f_{n}(x) = x^{(n)}(0)$ is continuous on $C^{\infty }[-1,1]$ for every $n = 0,1,2, \ldots$

Prove that every linear continuous functional on the space $s$ has the form $f\left(\left(x_{1}, x_{2}, \ldots \right)\right) = \sum_{n = 1}^{\infty } x_{n} y_{n}$, where $y = \left(y_{1}, y_{2}, \ldots \right) \in c_{00}$.

Prove that every linear continuous functional on the space $C(\mathbb {R})$ has the form $f(x) = \int_{\mathbb {R}} x(t) d y(t)$, where $y \in B V(\mathbb {R})$, and the measure $d y$ has compact support (i.e., there exists $A = A(y) > 0$ such that $y(t) \equiv \operatorname {const}_{1}$ for $t > A$ and $y(t) \equiv \operatorname {const}_{2}$ for $t < -A$).

Prove that every linear continuous functional on the space $C_{e}[0,1]$ has the form $f(x) = \sum_{k = 1}^{n} \lambda_{k} x\left(\alpha_{k}\right)$, where $n \in \mathbb {N}$, $\lambda_{k} \in \mathbb {R}, \alpha_{k} \in [0,1](k = 1, \ldots , n)$.

Prove that in a polynormed space with a countable family of seminorms, every sequentially continuous linear functional is continuous.

Prove that in the space $C_{\mathrm{e}}[0,1]$ every sequentially continuous linear functional is continuous.

Let the space $C_{0}^{1}(\mathbb {R})$ consist of continuously differentiable functions $x: \mathbb {R} \rightarrow \mathbb {R}$, each of which vanishes outside some (its own) segment. Suppose on this space the system of seminorms $\left\{ p_{g}: g \in C(\mathbb {R})\right\} \cup \left\{ P\right\}$ is introduced, where $p_{g}(x) = \sup \left\{ \left|g(t) x(t)\right|: t \in \mathbb {R}\right\} , P(x) = \sup \left\{ \left|x^{\prime }(t)\right|: t \in \mathbb {R}\right\}$. Prove that

Prove that in the following spaces there is no continuous linear functional other than the zero one:

Let $(X,\left\{ p_{\alpha }\right\}_{\alpha \in A})$ and $(Y,\left\{ p_{\gamma }\right\}_{\gamma \in G})$ be polynormed spaces, $T: X \rightarrow Y$ a linear map. Prove that the following conditions are equivalent:

1. $T$ is continuous at every point of the space $X$;

2. $T$ is continuous at zero;

3. for every seminorm $p_{\gamma }, \gamma \in G$, there exist indices $\alpha_{1}, \ldots \alpha_{n} \in A$ and a constant $C > 0$ (depending on $\gamma$) such that for every $x \in X$ the inequality $p_{\gamma }(T x) \leqslant C \max \left\{ p_{\alpha_{1}}(x), \ldots , p_{\alpha_{n}}(x)\right\}$ holds.

Prove that the differentiation operator $(A f)(z) = f^{\prime }(z)$ in the space $\mathcal{A}(D)$, where $D \subset \mathbb {C}$ is an arbitrary domain, is continuous.

Prove that the differentiation operator $(A x)(t) = x^{\prime }(t)$ is continuous in the space $C^{\infty }[0,1]$. Find its image and kernel.

Prove that the operator of multiplication by an infinitely differentiable function $(A x)(t) = a(t) x(t)$ is continuous in the space $C^{\infty }[0,1]$. Find its image and kernel in the cases

(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 $T \in \mathcal{M}$.