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.
6,814 questions
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If $\varphi(f)$ is riemann integrable for each $\varphi \in E^*$, then is $f$ riemann integrable?
Let $I=[0,1]$, $E$ be a banach space and $f:I \rightarrow E$ be a map.
Suppose that for every continuous functional $\varphi\in E^*$, the map $\varphi(f):I\rightarrow \mathbb{R}$ is riemann integrable....
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If $X$ is a Banach space with the finite tree property, can $\ell^1(\mathbb{Z})$ be finitely represented in $X$?
It is evident that if $\ell^1(\mathbb{Z})$ is finitely represented in $X$, then $X$ has the finite tree property. Indeed, $e_1,...,e_{2^n}$ in $\ell^1(\mathbb{Z})$ forms the $(n,2)$ part of a tree in $...
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There are two definition of resolvent set
Recently, when I studied the spectrum of an operator, I came across two definitions of the resolvent set. In Kreyszig’s book, the definition is as follows:
Let $X \ne {0}$ be a complex normed space ...
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Span of integer valued elements in $\ell_1(\mathbb{N})$
Consider the real Banach space $\ell_1(\mathbb{N})$, further denoted just $\ell_1$, consisting of all functions $x: \mathbb{N}\to\mathbb{R}$ such that $\|x\|:=\sum_{n=1}^\infty |x(n)|<\infty$, ...
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Reference request: Kadec-Klee property for $\ell^1$
I am looking for references (textbooks or articles) where it is shown that the Banach space $\ell^1$ has the Kadec–Klee property, i.e. that the weak topology and the norm topology coincide on the unit ...
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Orthogonal of the kernel [duplicate]
Let $H$ a Hilbert space and $T \in L(H)$ a linear self-adjoint operator, I cannot find the theorem on the internet that tells me that $\ker(T)^{\bot} = \overline{\operatorname{Ran}(T^*)}$.
I just want ...
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Definition of $C^{1}$ on closure of open set
Let $X$ and $Y$ be Banach spaces. Let $U$ be an open subset of $X$ and $f:A\rightarrow Y$ be a function with $U\subseteq A\subseteq\overline{U}$. Then we have two possible definitions of $f\in C^{1}(A,...
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$BV[0,1]$ with the norm $\lVert \cdot \rVert_{BV}$ is complete.
On $BV[0,1]$ we can introduce the following norm: $$\lVert f \rVert_{BV} := \lVert f \rVert_{\infty} + V(f)$$ [Here, $V$ is the total variation.] I want to show that this normed space is a Banach ...
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What’s the geometric intuition behind the Milman-Pettis theorem (uniform convexity ⇒ reflexivity)?
The Milman-Pettis theorem says that every uniformly convex Banach space is reflexive.
I’ve read and understood the proof, but I’m struggling with the motivation. The statement feels mysterious: ...
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A proof of Lindenstrauss
I am trying to understand an argument by Lindestrauss in his paper LIPSCHITZ IMAGE OF A MEASURE-NULL SET
CAN HAVE A NULL COMPLEMENT* and there is one passage I do not understand. I believe that he ...
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Context on the Arzelà–Ascoli theorem
I am doing an expository presentation on the theorem, and I understand the proof. What I fail to understand is, why is this theorem important? Why does any sequence $\{f_n\}$ containing a uniformly ...
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Sequence of vector units $x_1, x_2, \ldots$ in a normed space $X$ such that they are all linearly independent and their distances are $>1$.
I'm on my first semester of the Functional Analysis course that we have in my university. I've been stuck on this particular problem our professor presented to us not long ago for a while now:
Let $...
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The algebraic and analytic embedding of the disk algebra $A(\mathbb{D})$ in $BH(\mathbb{D})$
**My apologies in advance if this question is elementary, the reason I fear that it may be trivial is that I do not know some concrete examples of elements of $BH(\mathbb{D})\setminus A(\...
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On the completeness of $L_p$
There is this thing regarding this topic that confuses me, and I am afraid I am wrong about it, because as you know math has a lot of hidden details. Let $(X,\Sigma,\mu)$ be a measure space, $\mu(E)=0$...
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For normed vector space, can different norms make sequence to different limits? [duplicate]
A vector space can has different norms. A sequence can converge in a norm, but diverge in another norm. However, can there be:
A vector space $V$
Two norms $\left\| \cdot \right\|_1$, $\left\| \cdot \...
