# 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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### Weak convergence in $L^p$

My question is probably very basic, sorry about that.
Let $\{f_i\},\{g_i\}$ be two sequences converging to 0 weakly in $L^p[0,1]$ for any $p<\infty$. Can one conclude that $\int_0^1f_i(x)g_i(x) dx\...

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### $p$-nuclear operators from $C(K)$ to $L_{p}$

Let us recall that an operator $T$ from a Banach space $X$ to a Banach space $Y$ is called $p$-nuclear if $T$ can be written as $$T=\sum_{n=1}^{\infty}x^{*}_{n}\otimes y_{n},$$
where $\|(x^{*}_{i})_{i=...

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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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### Supporting Hyperplane Theorem in Lp Spaces

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 \...

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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 ...

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### 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 $E^n$.
...

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### 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?