Linear topological spaces

Suppose in a linear topological space the set $A$ is open, and the set $B$ arbitrary. Prove that the set $A+B$ is open.

Suppose in a linear topological space the sets $A$ and $B$ are closed linear subspaces. Is it true that $A+B$ is also a closed subspace?

Suppose in a linear topological space the set $A$ is closed, and the set $B$ is compact. Is it true that the set $A+B$ is closed?

Suppose in a linear topological space $X$ the sets $A$ and $B$ are compact. Will the set $C = A+B$ be compact?

Suppose in a linear topological space the set $A$ is open, and the set $B$ closed. Prove that for any scalar $\lambda$ the set $\lambda A$ is open, and the set $\lambda B$ is closed.

In a normed space, describe some local base $\mathcal{B}$ of neighborhoods of zero.

Give an example of a topology on a linear space with respect to which addition is continuous, while multiplication by a scalar is discontinuous.

Give an example of a topology on a linear space with respect to which addition is discontinuous, while multiplication by a scalar is continuous.

Prove that a linear topological space satisfies the third separation axiom $T_{3}$.

Prove that a linear topological space is separated (i.e., satisfies the first separation axiom $T_{1}$) if and only if the intersection of all neighborhoods of zero consists only of zero. In particular, every locally bounded linear topological space is separated.

Prove that on a finite-dimensional linear space $X$ there is only one topology (up to equivalence) with respect to which $X$ is a separated linear topological space.

Let $X$ be a linear space. Prove that among all separated topologies making $X$ a locally convex linear topological space there exists a strongest topology (it is called the kernel-convex topology). Prove that this topology can equivalently be defined as follows: the base of neighborhoods of zero consists of all convex sets containing zero and intersecting every line through zero in an interval of positive length.

Prove Theorem 14.1.

Theorem 14.1 (G. Birkhoff, S. Kakutani, 1936): A Hausdorff linear topological space $X$ is metrizable if and only if it has a countable local base of neighborhoods of zero.

Prove Theorem 14.2.

Theorem 14.2 (A. N. Kolmogorov, 1939): A Hausdorff linear topological space $X$ is normable if and only if it is locally convex and locally bounded.

Find the extreme points of the unit ball in the spaces

Find the extreme points of the unit ball in the space $BV_{0}[0,1]$ over the field $\mathbb {C}$.

Prove that among the complex spaces $l_{1}, l_{2}, l_{\infty }, c_{0}$, $c, L_{1}[0,1], L_{\infty }[0,1], C[0,1], B V[0,1], B C(\mathbb {R}), C_{0}(\mathbb {R})$ there is not a single pair of isometrically isomorphic spaces.

Prove that there exists an isometric embedding of the space $l_{\infty }$ into the space $L_{\infty }[0,1]$ and an isometric embedding of $L_{\infty }[0,1]$ into $l_{\infty }$ (although the spaces $L_{\infty }[0,1]$ and $l_{\infty }$ themselves, according to the previous problem, are not isometrically isomorphic).

Let $l_{p}, p \in (0,1)$, be the complex or real linear space of infinite sequences $(x_{1}, x_{2}, \ldots )$ for which the series $\sum_{k = 1}^{\infty }\left|x_{k}\right|^{p}$ converges. Define a base of neighborhoods of zero $\mathcal{B} = \left\{ G_{r}\right\}_{r > 0}$, where $G_{r} = \left\{ x: \sum_{k = 1}^{\infty }\left|x_{k}\right|^{p} < r\right\}$. Prove that this base defines a linear topological space that is not locally convex.

Verify whether a linear space with the discrete topology (this is the topology in which all sets are open) is a linear topological space.

Consider the space of equivalence classes of Lebesgue-measurable functions on $[0,1]$ with the metric

Prove that this is an $F$-space, which is locally bounded (but not locally convex — see problem 14.89) and separable. Prove that a sequence of measurable functions converges in the metric of this space if and only if it converges in Lebesgue measure on [0,1].