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# Questions tagged [banach-spaces]

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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### Compact space $K$ without convergent sequence while $c_0$ is complemented in $C(K)$

Let $K$ be a compact Hausdorff infinite topological space and $C(K)$ the Banach space of continuous functions from $K$ in $\mathbb{R}$ with sup norm. It is known that $c_0$ is complemented in $C(K)$ ...
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### Continuous Left-inverse of Dirac Lipschitz-Free Space

Let $X$ be a separable pointed metric space and let $AE(X)$ denote the corresponding Lipschitz-Free (or Arens-Eells space) over $X$. The point-evaluation map $\delta:X\mapsto AE(X)$ is injective ...
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### What imaginations of Lebesgue spaces or other Banach spaces do people intuitively share?

At several occasions I heared people discussing about the ?colors“ of Lebesgue spaces $L^p$: $L^2$ is red, $L^1$ is white, $L^\infty$ is black, and the other $L^p$ are blue or violett. Of course this ...
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### Are proper subspaces of Banach spaces which are isomorphic to the ambient Banach space necessarily complemented?

I had the following little question pop up, but I cannot seem to find any reference to it. Let $X$ be a Banach space and $E\subseteq X$ a proper subspace with $E$ isomorphic to $X$ itself. Is the ...
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### Verbal description, or terminology, for the ${\mathcal L}_p$-spaces of Lindenstrauss and Pelczynski

This question is intended for Banach-space specialists and so I will not repeat all the definitions here. My aim is to find out how the Banach space community refers to such spaces in discussions, and ...
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### subspace topology and strong topology

Suppose $X$ is a locally convex space and $Y$ is a subspace of the strong dual of $X$, is the induced topology on Y equivalent to the strong topology $b(Y,Y')$ on $Y$? If this is not correct, then on ...
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### Continuity of a differential of a Banach-valued holomorphic map

Originally posted on MSE. Let $U$ be an open set in $\mathbb{C}^{n}$ let $F$ be a Banach space (in my case even a dual Banach space), and let $\varphi:U\to F$ be a holomorphic map. I seem to be able ...
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Take C to be a closed convex set in $l^p$ (the space of sequences equipped with the $p$-norm where p>1) such that: i) 0 is in C ii) C is strictly larger than 0 iii) $C \cap -C =\{0\}$ iv) $C \... 0answers 131 views ### About an argument in the paper “Commutators on$\ell_\infty$” by Dosev and Johnson In the paper "Commutators on$\ell_\infty$" by Dosev and Johnson, in Lemma 4.2 Cas II, the authors have said that "There exists a normalized bock basis$\{u_i\}$of$\{x_i\}$and a normalized block ... 0answers 120 views ### Homeomorphism between$L^p$-spaces on metric spaces and$L^p$-spaces on Euclidean space Let$(X,d_X,\mu)$and$(Y,d_Y)$be complete metric spaces and$\mu$be doubling; moreover suppose that$(X,d_X)$is homeomorphic to a Euclidean space$E^D$and$(Y,d_Y)$is homeomorphic to$E^n$. ... 2answers 128 views ### Largest ideal in bounded linear maps on Schatten-$p$class Let$1\leq p<\infty.$Denote$S_p(\ell_2)$be the set of all compact operator$x$on$\ell_2$such that$Tr(|x|^p)<\infty.$Define$\|x\|_{S_p(\ell_2)}:=Tr(|x|^p)^{\frac{1}{p}}.$This makes$S_p(...
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### Question about pointwise convergence of operators

Consider two Banach spaces $E,F$ and a net $T_\alpha : E \to F$ of continuous operators. I know that for each $x \in E$ the net $T_\alpha (x)$ is convergent in $F$ and it is easy to show that the ...
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### Weaky compact subset of Banach space with separable predual

Let $X$ be a Banach space and $S\subseteq X$ be a subspace such that the unit sphere of $S$ is weakly compact. If $Y^*=X$ for some separable Banach space $Y,$ is it true that $S$ is separable?

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