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The weak topologies of the spaces $\ell_1$ and $L_1$ are not homeomorphic because of the following Theorem. Assume that $X,Y$ are two Banach spaces whose weak topologies are homeomorphic. If $X$ has S …

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 …

By Theorem 1.6 in the book "Geometric Nonlinear Functional Analysis" by Benyamini and Lindenstrauss, the Banach space $C[0,1]$ is a Lipschitz retract of the Banach space $\ell_\infty[0,1]$. Unfortunat …

It seems that the quostion about the skeletal property of $\psi$ has negative answer. Let us recall that a map $f:X\to Y$ between topological spaces is skeletal if for any nonempty open set $U\subsete …

Let $\tau$ be the weak topology on the Banach space $\ell_1$. It is known that each weakly convergent sequence in $\ell_1$ is norm convergent (i.e., $\ell_1$ has the Shur property). This property impl …

$\DeclareMathOperator\cov{cov}\newcommand\A{\text A}$A subset $A$ of a group $G$ is called affine if $A=xHy$ for some subgroup $H\subseteq G$ and some $x,y\in G$. Given a non-discrete topological grou …

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 …

The answer here is negative. Given any infinite-dimensional Banach space $X$, fix any non-zero linear continuous functional $f:X\to\mathbb R$ and fix any vector $x_1\in f^{-1}(1)$. Take any dense $G_ …

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 …

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 Banach space $X\ne \{0\}$, let $\mathrm{cov}_H(X)$ be the smallest number of hyperplanes covering $X$. By a hyperplane in a Banach space I understand any closed affine subspace of codimension …

Let $B=-B$ be a nowhere dense bounded closed convex set in the Hilbert space $\ell_2$ such that the linear hull of $B$ is dense in $\ell_2$. Question. Is there a non-compact linear bounded operato …

Let us call a series $\sum_n x_n$ in a Banach space "good" if there exists a permutation $\sigma:\mathbb N\to\mathbb N$ such that the rearranged series $\sum_n x_{\sigma(n)}$ converges. Find a simpl …

An example of a non-separable Banach space $X$ with $|B(X)|=\mathfrak c$ is any non-separable Banach space $X$ whose dual $X^*$ is $w^*$-separable and has cardinality $|X^*|=\mathfrak c$. This foll …