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For a linear space $X$ with a Hamel basis $H$ the largest vector topology seems to coincide with the topology of free linear topological space over the discrete space $H$. If this is true, then we can …

A bounded subset $B$ of the dual $X^*$ of a Banach space $X$ is called norming if the formula $\|x\|:=\sup\{|x^*(x)|:x^*\in B\}$ determines an equivalent norm on $X$. Observe that the sequence $(e_n^ …

Problem 1. Characterize compact Hausdorff spaces $K$ for which the Banach space $C(K)$ of continuous real-valued functions embeds into the Banach space $c_0$. Since $c_0$ has separable dual, such …

For a locally convex space $E$ let $E_\beta$ be the space $E$ endowed with the locally convex topology $\beta(E,E')$ whose neighborhood base at zero consists of barrels, i.e., closed absolutely convex …

For two functions $x,y:\omega\to\mathbb R$ let $xy:\omega\to\mathbb R$, $xy:n\mapsto x(n)y(n)$, be their pointwise product. It is clear that for any elements $x,y\in\ell_2$ their pointwise product $xy …

Assume that a sequence $(x_n)_{n\in\omega}$ of points of a locally convex topological vector space converges to zero. Is it always possible to find increasing number sequences $(n_k)_{k\in\omega}$ and …

Oh, sorry! I posed this question too quickly. A simple example is the dual space $\ell_1=c_0^*$ to the Banach space $c_0$, endpowed with the weak$^*$ topology. The sequence $(e^*_n)_{n\in\omega}$ of …

Volodymyr Kadets kindly informed me that the answer to this problem is affirmative. His argument easily generalizes to prove the following Theorem. For any $p\in[1,\infty)$, any unconditionally conve …

Let $A$ be a linear analytic subspace of a Polish vector space $X$. Using Piccard-Pettis Theorem, it is easy to prove that $A$ has uncountable codimension in $X$. If $A$ is of type $F_\sigma$, then ap …

I need a reference to the following (known?) Fact. Every topological vector space $X$ over the field of real numbers is topologically isomorphic to a linear subspace of the Tychonoff product of li …

Definition. Let us define a Banach space $X$ to be co-Sobczyk if every linear bounded operator $T:Z\to c_0$ defined on a separable subspace $Z$ of $X$ extends to a bounded operator $\bar T:X\to c_0 …

For a function $x:\omega\to\mathbb R$ let $|x|$ denote the function $|x|:\omega\to[0,\infty)$, $|x|:n\mapsto|x(n)|$. It is well-know that a series $\sum_{n\in\omega}r_n$ of real numbers converges unco …

Given a real number $c\ge 1$ let us say that a metric space $X$ is a $c$-Lipschitz comp-extensor if each Lipschitz self-map $f:K\to K$ of a compact subset $K\subset X$ extends to a Lipschitz self-map …

The answer to the main question is negative: Consider the compact subset $X=[0,1]\cup \{2\}$ of the real line and let $Y=\{2\}$ be a singleton in $X$. In the function space $C(X)$ consider the closed …

Let $(X,\|\cdot\|)$ be a 2-dimensional real Banach space and $S=\{x\in X:\|x\|=1\}$ be its unit sphere. Assume that $S$ is smooth in the sense that for any $y\in S$ there exists a unique functional $y …