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It is an easy undergraduate exercise to show that (finite) direct sums are preserved under dualisation. Thus, it is natural to ask if we the following holds: is it true that if $X$ is a subspace of $ …

When browsing the literature, I have found the following theorem of E. Tokarev: Let $X$ be a Banach lattice with weakly sequentially complete dual space. Then for any Banach space $Y$, every uncondit …

Consider the space $E_n = C^1([0,1]^n)$ of continuously differentiable functions with the usual norm $$\max\{ \|f\|_\infty, \|f^\prime_{x_1}\|_\infty, \ldots, \|f^\prime_{x_n}\|_\infty\}.$$ making it …

Consider the Banach space $L_1(\mathbb R^n, \mathbb R^n)$ of integrable vector fields $(n>1$) together with its subspace $N$ formed by those vectors fields whose divergence (computed in the distributi …

Suppose that we have a Banach space $X$ together with some locally convex Hausdorff topology on $X$, weaker than the one given by the norm, which makes the unit ball of $X$ compact. Is $X$ (Banach-spa …

Recall that a biorthogonal system $\{(x_i, x^*_i)\colon i\in I\}$ in a Banach space $E$ is a M-basis if $\{x_i\}_{i\in I}$ is linearly dense in $E$ and $\{x_i^*\}_{i\in I}$ separates points. Let me re …

Let $\omega_1$ be smallest uncountable ordinal. I am trying to understand the possible "large" subspaces of $C[0,\omega_1]$, namely those which are isomorphic to the whole space. Therefore I have the …

There is probably an embarrassingly simple counter-example to my question but I couldn't figure it out myself. Let me give it a try here. A Banach space $X$ is Grothendieck if weak*-convergent sequen …

Let $X$ be a set of cardinality $\aleph_1$ and consider the Banach space $\ell_\infty^c(X)$ of all scalar-valued bounded functions on $X$ which are non-zero only for countably many elements of $X$ end …

This must be surely known but I couldn't locate this problem in the literature. It popped out in a priori unrelated approximation problem but if true, would help me greatly. Let $p\in (1,\infty)$. I …

The question, already phrased in the title, looks like a classical problem from Banach space theory from the 1970s. Hence, my question is more of a reference request in its nature. Can every separ …

Are the weak topologies of $\ell_1$ and $L_1$ homeomorphic? Strangely may it sound, the question seeks contrasts between norm and weak topologies of Banach spaces from the non-linear point of view. …

Let $\ell_p^n$ be the $n$-dimensional real or complex $\ell_p$-space and let $\mathscr{B}(\ell_p^n)$ be the space of matrices on $\ell_p^n$ endowed with the operator norm. I am looking for any referen …

The following inequality is an elementary exercise in convexity: let $x,y$ be non-zero vectors in a normed space with $\|x\|, \|y\|\leqslant 1$. Suppose that $\|x-y\| \geqslant 1$. Then $$\left\|\fra …

Can one construct in ZFC a Banach space of density character $\omega_1$ that does not have an equivalent strictly convex norm? Maybe one may apply some kind of a Löwenheim–Skolem-type argument to …