# Tagged Questions

A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

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### About a partial converse to the Banach-Steinhaus Theorem

I've been reading the GTM text Topics in Banach Space Theory by Albaic and Kalton. In the appendix, it states the following partial converse to the Banach-Steinhaus theorem: Let $\{S_n\}$ be a ...
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### Examples of compact sets that are infinite dimensional and not bounded

In an infinite dimensional Banach space, does a compact subset have to be finite dimensional? I know it cannot contain any infinite dimensional balls, if this mean it has to be finite dimensional, ...
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### Something is wrong with this argument (Lorentz and Rosenthal-Woo sequence spaces)

Fix once and for all $1<p<\infty$. Throughout, $w=(w_n)_{n=1}^\infty$ will denote a sequence of positive real weights satisfying 1=w_1\geq w_2\geq w_3\geq\cdots>0\;\;\;\text{ ...
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### The set of w*-continuous operators is closed for the weak* topology?

Let $X$ be a dual Banach space, i.e. $X=(X_*)^*$ for some Banach space $X_*$. Consider the weak* topology of $B(X)$, i.e. the topology of pointwise convergence on $X$ endowed with the $\sigma(X,X_*)$-...
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### Does the norm of an operator on a $w^{*}$ dense subspace determine its norm?

Let $X$ be a (separable) Banach space, $T:X^{*}\to X^{*}$ a bounded operator, and $Y\subset X^{*}$ a norm closed, $w^{*}$ dense subspace of $X^{*}$. Is it true that $\|T\|=\|T_{|Y}\|$?
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### Is there a Banach space which is not isometrically isomorphic to $l^p$?

I know that every Hilbert space is isometrically isomorphic to $l^2(\beta)$ where $\beta$ is a Hilbert basis for that space. Do Banach spaces have the similar property? That is, is every Banach space ...
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### If $\varphi \in E'$ and $A$ is convex and open then $\varphi (A)$ is an open interval

Let $E$ be a real normed space and $\varphi \in E'$, $\varphi \neq 0$. Suppose that $A \subset E$ is an open convex not empty subset. Show that $\varphi(A)$ is an open interval. Since $A$ is ...
Given the Banach space $X:=\mathcal{C}([0,1]\cup[2,3])$. I remember I've seen a beautiful example of a non-closable operator whose graph is dense. It involved exploiting Stone-Weierstraß for a ...
Given two Banach spaces $X$ and $Y$. (More generally locally convex spaces) Regard a closed subspace $U\subseteq X$. Does every bounded operator extend: T\in\mathcal{B}(U,Y)\implies T_E\in\mathcal{...