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For question (1), weakly Lindelöf determined spaces (WLD spaces, for short) does it. A WLD space is characterised by the dual ball, modulo the weak-star topology, being a Corson compactum (i.e., for s …

The answer to both Q1 (in the context given) and Q2 is yes. Regarding the weak closure of an arbitrary set of linear functional being linear that appears to me to be sequential weak-star convergence — …

Let $T\colon\mathcal{X}\to\mathcal{X}$ be a compact operator with a dense range on an infinite-dimensional separable Banach space $\mathcal{X}$ $.^{*}$ Then $K:=\overline{T(B_1(0))}$, where $B_1(0)$ i …

I want to believe that this has an easy answer, but I’ve never considered it before and can’t seem to answer it now either. Does every infinite-dimensional Banach space admit a locally convex vector …

Let $\mathcal{X}$ be a decomposable Banach space (i.e. a topological direct sum of infinite-dimensional subspaces, say $\mathcal{X}=\mathcal{A}\oplus\mathcal{B}$). Can one always obtain another decom …

This is impossible unless the following holds: $q=2$ and either $p=2$ or $r=0$. The above exception is due to the fact that it implies $\frac{1}{1+r}\eta(\cdot)=\|\cdot\|_q=\|\cdot\|_2$, for which t …

While attempting to answer this question, I recalled that the term “bounded” can be pretty confusing in normed vector spaces if not clarified; in general they all take the form $$\|Tx\|\le L\|x\|+M\,, …

The property you indicate is known as (strict) Opial’s Property (see https://en.m.wikipedia.org/wiki/Opial_property). It fails generally in reflexive spaces; in fact, it fails generally even for unifo …

So my question is straightforward. Let $\mathfrak{X}$ be a (complex, if necessary) Banach space and $K\colon\mathfrak{X}\to\mathfrak{X}$ a nonzero compact operator. Denote by $\mathcal{C}(K)$ the comm …

Simply observe that $$\|x\|_{a,b}=\|x_1a+x_2b\|_1\,.$$ Thus, by orthogonality of $a,b$ and the easily-derived inequality $\|y\|_2\le\|y\|_1\le\sqrt{n}\|y\|_2$ for any $y\in\mathbb{R}^n$, we have \begi …