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Let $X$ be a Banach Space and let $Y$ be a closed subspace of $X^{**}$ such that $X\bigcap Y=0$. Let $P$ be the quotient map from $X^{**}$ onto $X^{**}/ Y$. I need to prove or refute that $P\left|_{X} …

Consider the following properties of a Banach space: the intersection of any support hyperplane with the unit sphere is (S) a singleton (this is the strict convexity); (SF) finite-dimensional set …

Let $E$ be a Banach space. Let $F$ be the collection of all $f\in E^*$ such that $\left<f,e_n\right>\to 0$, for every normalized basic sequence $\{e_n\}$. It is easy to see that $F$ is a closed subspa …

This answer is supplementary to the one of Bill Johnson, to fill in some details. A sequence $\{e_n\}$ in a Banach space $E$ is called a basic sequence of type P* if (among other equivalent definitio …

Let $F$ be a Banach space with the closed unit ball $B$. Let $E\subset F^*$ be a total subspace such that $B$ is complete with respect to the norm $|||f|||=\sup \limits_{e\in E,~e\ne 0} \frac{|\left<f …

Let $E$ be a Banach space. It is known that if for any equivalent norm on $E^*$ the closed unit ball of $E^*$ is weakly* closed, then $E$ is reflexive (a very short proof is in the book by Fabian, Ha …

Originally I meant to ask this question here, but got confused and ended up asking another question, which had some mathematical meaning, but was not what I vaguely had in mind. Let me restate the mo …

Originally asked on MSE. Let $T$ be a linear map from a normed space $E$ into a Banach space $F$. Let $D\subset \overline{B}_{F^{\ast}}$ be norming, i.e., there is $r>0$ such that $\sup\limits_{v\in …

This is a complement of my previous question about the sequence spaces (I'm afraid, there will be a third part). Let $\rho:[0,+\infty)^{\mathbb{N}}\to[0,+\infty] $ satisfy the following properties: …

Let $\rho:[0,+\infty)^{\mathbb{N}}\to[0,+\infty] $ satisfy the following properties: $\rho(\lambda u)=\lambda\rho( u)$, for every $u\in [0,+\infty)^{\mathbb{N}}$ and $\lambda\ge 0$; $\rho(u+v)\le \r …

Recall that a Hausdorf topological space $X$ is called compactly generated if any set whose intersections with compacts are compact is closed. Locally compact and first countable spaces are compactly …

Let $E$ be a Banach space, let $e_{n}\in E$ and $g_{n}\in E^{*}$ be biorthogonal basic sequences (i.e. $\left<e_n,g_m\right>=\delta_{mn}$ ). Moreover, both of these sequences are weakly null. (note th …

I think I've found a counterexample in the literature. I would really appreciate if somebody verified that I didn't get confused about the terminology of Orlicz spaces. Recall that a Banach space $E$ …

Let $X$ be a compact metric space, and let $B$ be a convex balanced bounded set in $C(X)$ such that for every $x\in X$ there is $f\in B$ with $f(x)\ne 0$. Let $M=\{u\in C(X),~ uf\in B,~\forall f\in B …

The title says it all: Let $E$ be a Banach lattice, which is isomorphic to a Hilbert space (as normed spaces). Is there an equivalent Hilbert norm on $E$, which still makes it a Banach lattice with r …