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Mathematics Stack Exchange

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.

A Banach space, named after Stefan Banach (1892–1945) is a complete normed vector space: a (real or complex) vector space equipped with a norm such that every Cauchy sequence converges. For instance, $\mathbb{R}^n$ and $\mathbb{C}^n$, equipped with the usual norm (or, for that matter, any norm) is a Banach space. Another example is the space $\ell^1$ of all absolutely convergent series of real or complex numbers, equipped with the norm $\left\|\sum_{n=0}^\infty x_n\right\|=\sum_{n=0}^\infty|x_n|$.

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Sum of Banach Spaces is complete

Let $A_1, A_2,...$ be a sequence of Banach spaces with $\|\cdot\|_n$ denoting the norm on $A_n$. Let $p\in[1,\infty)$ and $$\sum\limits_pA_n:=\{(a_n)_{n=1}^{\infty}| a_n\in A_n \text{ and } \sum\limits_{n=1}^\infty\|a_n\|_n^p<\infty\}.$$ Show that…
mi986
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What is an example of nested bounded convex and closet sequence of set in a Banach space with empty intersection?

What is an example of nested bounded convex and closet sequence of set in a Banach space with empty intersection? I cannot imagine an example of that. Thanks.
José
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Linear functionals on Banach spaces

The following is a homework problem (I have solved the majority of it, but need help with the last part) Suppose $f$ : $V \rightarrow \mathbb{F}$ is a non-zero continuous linear functional on a Banach space $V$ over $\mathbb{F}$. I. Show that W =…
rt93
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Showing projection is continuous if and only if kernel is closed

I have a linear map $P$ on a Banach space, $X$, with $P^2 = P$ and I'm trying to show that $P$ is continuous if and only if $\ker(P)$ and $\ker(I-P)$ are closed. One direction is straight forward but I'm struggling with the other. I noticed we can…
Wooster
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A question in Banach space

Given $A\in\Bbb R^{m\times n}$, define $$\gamma_2(A)=\min_{XY=A}\max_{i,j}\|x_i\|_{\ell_2}\|y_j\|_{\ell_2}$$ where rows of $X$,columns of $Y$ given by $\{x_i\}_{i=1}^m,\{y_j\}_{j=1}^n$ respectively. Where could I find a reference for fact that…
Turbo
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Distortion and Norm Stabilization

I'm curious about the Distortion Problem in Banach space theory and its relation with norm stabilization. I found that if $(X, \| \cdot \|)$ is an infinite-dimensional separable Banach space, then $X$ does not contain a distortable subspace iff…
ragrigg
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Let $V$ be a normed vector space over $\mathbb{C}$, is there an inner product structure on $V$ such that the two spaces have the same topology.

Let $V$ be a complete normed vector space over $\mathbb{C}$. Let $\tau_1$ be the topology induced by its norm. Is there an inner product structure on $V$ such that the topology induced by the norm of the inner product is the same as $\tau_1$ ? I…
Amr
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Understanding the relation of weak and weak star toplogy

I'm working with Eberlein- Smulian Theorem fromm the book "Topics in Banach Space Theory". During the proof I have seen that there is used a lot the concept of weak topology and weak star topology. For a given Banach space X according to the…
Kristina Dedndreaj
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Do I have a Banach space given this wacky norm?

I have a normed space (I'll denote it $C^2[a;b]$) which consists of continuous real functions whose first and second derivatives are also continuous in interval $[a;b]$. $\forall x,y \in C^2[a;b]$ and $\forall \lambda \in \mathbb{R}$ the norm is…
Pranasas
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Let X be a Banach space, and x,y ∈ X. Show that if ℓ(x) = ℓ(y) for every ℓ ∈ X′, then x = y.

Not too sure where to start. Seems like it should be an easy adaptation of a theorem, but have't been able to find it in my notes. Thank you!
Kristaps
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Prove $B_{m+n}=B_m * B_n$ if $B_k$ is the sigma-algebra of all Borel sets in $\mathbb R^k$

Let $B_k$ be the sigma-algebra of all Borel sets in $\mathbb R^k$. How can we prove that $B_{m+n}=B_m * B_n$? I am a beginner in analysis, hope to seek help here.
Lynette
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Is the Euclidean norm the unique norm which turns $\mathbb{R}^n$ into a Banach space?

See the title. Essentially I'm wondering if the answer is no, does there exist two inequivalent notions of calculus on $\mathbb{R}^n$ obtained via the Frechet derivative for the two resulting (inequivalent) Banach Spaces.
Okazaki
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Is $(l^2, \|.\|_2')$ Banach space?

Let $l^2=\left\{(x_n)_n|\, \sum\limits_{n=1}^{\infty}|x_n|^2<\infty\right\}$ with norm $$\|x\|_2'=\left(\sum\limits_{n=1}^{\infty}\frac{n^2}{n^2+1}|x_n|^2\right)^{1/2}$$ So, is it Banach space? I know how to done it with norm $\|.\|_2$ but now I'm…
jagodicans
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Is "$A \subset X^*$ separates points of $X$ iff. $\overline{\operatorname{conv}}^{w^*}(A) = X^*$" a correct statement?

Is "$A \subset X^*$ separates points of $X$ iff. $\overline{\operatorname{conv}}^{w^*}(A) = X^*$" a correct statement? I saw this statement in a question in a book but it didn't seem right to me, so I just want to make sure that I'm not missing…
Project Book
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Isometries on $L^p$ spaces and generalized inverses

In the paper Partial isometries and generalized inverses by Mbekhta, a bounded linear operator $T$ on a complex Banach space $X$ is defined to be a partial isometry if it is a contraction and it admits a generalized inverse that is also a…
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