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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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defining a topology by its compact sets

The goal. Let $X$ be a set endowed with Hausdorff topologies $\tau_w$ and $\tau_n$, such that $\tau_w\subseteq\tau_n$. Let $\mathscr{C}$ denote a family of subsets $A\subseteq X$, which satisfies ...
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If a map between separable Banach spaces has closed graph, does it have a point of continuity?

It is well known that the closed graph theorem does not directly extend to nonlinear maps: even for functions from $\mathbb{R}$ to $\mathbb{R}$, having closed graph does not imply continuity. But let'...
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Understanding Lang's Proof of Fubini's Theorem

This question concerns the proof of Theorem 8.4 (Fubini's Theorem part 1) on page 162 in Lang's real and functional analysis book. To understand the proof I need to give following background from the ...
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Random variables that span copies of $\ell_p$

Consider the coin-toss measure $\mu$ on $\{0,1\}^\mathbb{N}$. Within this framework it is easy to construct a sequence of independent, symmetric Bernoulli random variables. Indeed the point-evaluation ...
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Stable local minimizers of functions on a Banach space

Let $X$ be a Banach space and $f:X\rightarrow (-\infty,\infty]$ be a lower semicontinuous function. We are interested in some conceptions of local minimizer: We say that $\bar{x}\in X$ is a stable ...
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Density of Banach spaces

I am trying to understand the notion of density in the context of Banach spaces. The density of a topological space is the least cardinality of a dense subset. Thus, a separable Banach space has ...
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Dominated convergence theorem for Banach space

I'm trying to prove dominated convergence theorem for Banach space. Could you verify if my proof is fine or contains some subtle mistakes? Let $(f_n)$ be a sequence in $\mathcal L_0 (X, \mu, E)$. ...
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$j(X)$ weak*-dense in $X^{**}$, $ j$ is the canonical embedding

Let $X$ be a Banach space and consider the canonical embedding in it's bidual $X^{**}$, namely $j:X\to X^{**}, \; x\mapsto j(x)$, where $j(x)(x^*)=x^*(x)$ for $x^*\in X^*$. My question: Why is $j(X)$...
Normed space's user avatar
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renorm a Banach space to make an operator have spectral radius equal to norm

Let $X$ be an infinite-dimensional complex Banach space equipped with the norm $\lVert\cdot\rVert$, and let $T\in\mathcal{L}(X)$ a bounded linear operator on $X$. Let $r(T)$ denote the spectral ...
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Complemented Banach spaces.

Let $X$ be Banach space and $Y$ a closed subspace of $X$. Assume that there exist a closed "subset" $Z$ of $X$ with the properties: $Z\cap Y=\{0\}$ and every $x\in X$ can be written in a unique ...
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Spectrum of restriction to invariant subspace

Let $H$ be a separable Hilbert space and let $\mathcal{B}(H)$ denote the algebra of linear bounded operators on $H$. Let $T \in \mathcal{B}(H)$ and let $M$ be a non-trivial closed invariant subspace ...
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Dual Space of General $L^p$ Space Which Takes Values in Banach Space

Last week, our functional analysis course covered Riesz Representation theorem for $L^p(X,\mu),(1\leq p < \infty)$, namely, $(L^p(X,\mu))^* = L^q(X,\mu)$. And I was stuck with this homework problem ...
Edward Z. Miao's user avatar
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How to define orientation on infinite dimensional vector space

Let $\mathbb{V}$ be a real Banach space (if someone knows the answer for more arbitrary T.V.S. then great). Is there some concept of orientation that one could define on $\mathbb{V}$ that matches the ...
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Is the space of $C^k$ submanifolds a Banach manifold

Let $M$ and $N$ be (finite-dimensional) smooth manifolds without boundary. For simplicity, assume $M$ is compact. This post concerns spaces of "copies of $M$ in $N$". To the best of my knowledge, the ...
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Stability of point spectrum

Suppose $T$, $S$ are bounded operators on $l_2$, $a_n\to 0$ a sequence of complex numbers with the property that for any $n\in\mathbb{N}$, $T+a_nS$ has discrete spectrum and non-empty point spectrum....
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Areas of research / interest in Banach space theory today.

Whilst taking a class in functional analysis I couldn't help but feel Banach space theory was only ever taught as a natural stepping stone towards Hilbert space theory. As to the prominence and status ...
Jeremy Jeffrey James's user avatar
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A possible norm on a subspace of $C^\infty([0,1])$?

My question is related to this one: Take the vector space of infinitely differentiable functions on $[0,1]$. The standard norm of $C^k([0,1])$ is just the $\ell^1$-norm of the vector $(\|f\|_\infty, \|...
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Does the Closed Graph Theorem follow from Banach-Steinhaus?

Q: Is there a simple (but perhaps tricky or clever) proof of the Closed Graph Theorem (or the Open Mapping Theorem, or the result I call the Automatic Inverses Theorem below) from the Banach-Steinhaus ...
David C. Ullrich's user avatar
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Why should the open mapping theorem be expected?

Soft question alert. I want to know why to expect the open mapping theorem to be true. My thoughts: I know that one nice consequence of the OMT could be thought of as the universal property of ...
Elle Najt's user avatar
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Radon-Riesz & Kadec-Klee

Let us say that a normed vector space has the a) RR (Radon-Riesz) property if for any sequence, norm convergence is equivalent to weak convergence + convergence of norms. b) KK (Kadec-Klee) property ...
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Representation of Stochastic Integrals as Lebesgue/Bochner Integrals

Just as the Riemann–Stieltjes integral can be equivalently defined as a Lebesgue integral with the corresponding Lebesgue–Stieltjes measure, I am looking for the corresponding results for the ...
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Subsequences of a basic sequence

Suppose ($x_n$) is a basic sequence in a Banach space $X$, and $Y$ is a closed, infinite co-dimensional subspace of the closed span of $(x_n)$. Can we always find a subsequence ($y_n$) of ($x_n$) such ...
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Equivalence of definitions of $C^k(\overline U)$

let $U$ be an open set of $\mathbb{R}^n$, that contains at least some open set. In Evans book we find the definition $$C^k(\overline U)=\{f \in C^k(U): D^\alpha f \text{ is uniformly continuous on ...
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Integration for functions with values in a separable Banach space

Let $(X,\mathcal{M},\mu)$ be a measure space, $Y$ a separable Banach space, and $L_{Y}$ the space of all $(\mathcal{M},\mathcal{B}_{Y})$-measurable maps from $X$ to $Y$ (where $\mathcal{B}$ denotes ...
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Borderline case of interpolation of Banach spaces

Let $B \subset A$ be Banach spaces with a continuous embedding. Is the inequality $$ \|b\|_B \leq C \sup_{t > 0} \inf_{\tilde{b} \in B} \{ \|b - \tilde{b}\|_B + t \|\tilde{b}\|_A \} \quad \forall ...
user66081's user avatar
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Finite dimensional subspaces

Let $X$ be a complex Banach space of infinite dimension. Does there exist a finite dimensional subspace of $X$ of arbitrary (finite) dimension which is complemented by a projection of norm 1?
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Hadamard differentiability of function

Let $X$ and $Y$ be Banach spaces. Definition: A function $f:X\rightarrow Y$ is called Hadamard differentiable at $x\in X$ tangentially to $U\subseteq X$ iff $x\in U$ and there exists a continuous ...
Syd Amerikaner's user avatar
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Showing linear isometry

Define a norm on $\ell^1$ by $$\|x\|=(\|x\|_1^2+\|x\|_2^2)^{\frac{1}{2}},$$ where $\|.\|_p$ denotes the canonical norm on $\ell^p$. Then $\|.\|$ is equivalent to $\|.\|_1$. I want to show that a ...
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Luxemburg norm as argument of Young's function: $\Phi\left(\lVert f \rVert_{L^{\Phi}}\right)$

Let $\Phi$ be a Youngs's function, i.e. $$ \Phi(t) = \int_0^t \varphi(s) \,\mathrm d s$$ for some $\varphi$ satifying $\varphi:[0,\infty)\to[0,\infty]$ is increasing $\varphi$ is lower semi ...
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Showing that the equation $x_i - \sum_{j=1}^\infty a_{ij}x_j = b_i$ has a unique solution.

Exercise : Consider the infinite-dimensional system of equations : $$x_i - \sum_{j=1}^\infty a_{ij}x_j = b_i, \quad i=1,2,3,\dots$$ We suppose that $b=(b_1,b_2,\dots) \in \ell^\infty$ and that ...
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Comparaison of two version of Fractional sobolev spaces: what do we have $W^{s,p}(\mathbb{R}^{n})=H^{s,p}(\mathbb{R}^{n})$?

There are two version of Fractional sobolev spaces . Definition1: (Via Galiardo semi-norm) Let $1\leq p\leq +\infty$, $0<s<1$ and $\Omega\subseteq \mathbb{R}^n$ an open set. The fractional ...
Guy Fsone's user avatar
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Is every $C_0(K,X)$ space isomorphic to $C(L,X)$, for some compact $L$?

Let $K$ be a locally compact Hausdorff space and $X$ be a Banach space. Denote by $C_0(K,X)$ the Banach space of all continuous $X$-valued functions defined on $K$ that vanish at infinity, equipped ...
Vinícius Morelli's user avatar
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The dual of the Banach space $C(\Omega)$

It is well-known that the dual of the Banach space $C([0,1])$, i.e. the space of all continuous functions on the interval, is the space of all functions of bounded variation on the interval, $BV([0,1])...
Jeff Kenney's user avatar
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Weak*-complemented subspaces of $\ell_\infty$

Consider $\ell_\infty$ as $\ell_1^*$. Let $X$ be an infinite-dimensional complemented subspace of $\ell_\infty$ (in partiuclar, $X$ is isomorphic to $\ell_\infty$). Can we find a further subspace $Y\...
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Properties shared by equivalent norms.

I am interested in knowing about "geometric" properties shared by equivalent norms on a Banach space. Here I mean "geometric" as opposed to topological, and probably in particular with reference to ...
Elle Najt's user avatar
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6 votes
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Reference request for the fact

Does anyone know a reference to the paper or a textbook where this fact is proved $$ \mathcal{B}(\bigoplus_1 X_\alpha, Y)\cong_1 \bigoplus_\infty \mathcal{B}(X_\alpha, Y) $$ Most author are bored to ...
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Proving the $l_p$ space is complete.

I'm trying to prove $l_p$ spaces are complete. We have an $l_p$ space $W$. Let us take a cauchy sequence. There exists $N_0\in\Bbb{N}$ such that for $m,n>N_0$, $d(x^m,x^n)<\epsilon$. This ...
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Linear isomorphisms with dense graph

Is it true that for each infinite dimensional Banach space $X$ there exists a linear bijection $f: X \rightarrow X$ with a dense graph? A graph of $f$ it is the set $\Gamma(f):=\{(x, f(x)): x \in X \...
Richard's user avatar
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5 votes
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Property of vector-valued measure

Let $B$ be a Banach space, let $(X,\mathcal{A})$ be a measurable space, and let $\mu:\mathcal{A}\to B$ be a vector-valued measure of bounded variation. In general, if $B$ doesn't have the Radon-...
geodude's user avatar
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Identifying linear operator with a bilinear symmetric form using Theorem of Schwarz

We study the energy functional $E$ of the form $$E(v)=\frac{1}{2}a(v,v)+\int_{\Omega}F(x,v).$$ Let $V$ be a real Banach space with norm $||\cdot||_{V}$ and denote by $V^{'}$ the dual space. By $\...
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Application of Banach-Alaoglu theorem to extract convergent subsequence of currents

While reading about currents I came across the following lemma in Lectures on Geometric Measure Theory by Leon Simon on page 135: Lemma. If $\left\{T_j\right\}_{j\in\mathbb{N}}$ is a sequence of ...
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Sufficiency part For the existence of $\bar p(\cdot)$ $\in$ $W(p)$ for which $C([0,1]$) is closed subspace in $L^{\bar p(\cdot)}([0;1])$.

Handwriting this would be impossible, so I apologize. These are the definitions and theorems which we need for the proof of the theorem : BFS is defined as the Bannach Function Space. Let $W(p)$ ...
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An equivalent norm of $\|\cdot\|_\infty$

Let $X$ be a compact topological space and $(E, |\cdot|)$ a Banach space. Let $\mathcal C$ be the space of all continuous functions from $X$ to $E$. Let $\|\cdot\|_\infty$ be the supremum norm on $\...
Akira's user avatar
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Banach space is separable iff Borel $\sigma$-algebra coincides with $\sigma$-algebra generated by open balls

The question is essentially in the title. Let $X$ be a Banach space, $\mathcal B(X)$ be a Borel $\sigma$-algebra, $\Sigma$ be a $\sigma$-algebra generated by the collection of all open balls. Is it ...
George Stobbart's user avatar
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The category $\mathbf{Ban_1}$ equipped with a non-obvious functor $U_1: \mathbf{Ban_1} \to \mathbf{Set}$

Wikipedia says that For technical reasons, the category $\mathbf{Ban_1}$ of Banach spaces and linear contractions is often equipped not with the "obvious" forgetful functor but the functor $...
stoic-santiago's user avatar
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Is this a typo in Brezis's Ex 3.24?

I'm doing Ex 3.24 in Brezis's book of Functional Analysis. The purpose of this exercise is to sketch part of the proof of Theorem 3.29, i.e., if $E$ is a Banach space such that $B_{E}$ is metrizable ...
Analyst's user avatar
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$E$ is a Banach space for the norm $\| \cdot \|_1$ where $\|x\|_{1} := \|x\|_{E}+\|T x\|_{F}$

I'm reading a proof of closed graph theorem in textbook Functional Analysis, Sobolev Spaces and Partial Differential Equations. Let $E$ and $F$ be two Banach spaces. Let $T$ be a linear operator from ...
Akira's user avatar
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How to prove that the map $T \mapsto T^{-1}$ is continuous?

Let $E$ be a Banach space. Let $\mathcal L (E)$ denote the space of all bounded linear operators on $E$ and $\text {GL} (E)$ denote the space of all bounded linear operators on $E$ with bounded ...
Anil Bagchi.'s user avatar
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Minimum Solution Over Closed Ball of $H_0^1(\Omega)$

Let $\Omega\subset \mathbb{R}^n$ an open bounded domain. Let $\kappa:\Omega \to \mathbb{R}$ a continuous function which $\beta\leq \kappa(x)\leq M \quad \forall x \in \Omega$, where $0<\beta,M.$ ...
Pablo Herrera's user avatar
5 votes
1 answer
263 views

Sum of Banach spaces norm

Let $ \left ( X_{1},\left \| \right \|_{1} \right ) $,$ \left ( X_{2},\left \| \right \|_{2} \right ) $ two Banach spaces in the vector space X. How to prove that $\left \| x \right \|= \inf \left \...
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