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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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Interpolation of some Sobolev spaces

Let $X_0=L^2(0,1)$, $X_1=H^4(0,1)$, $X_2=H^4(0,1)\cap H^2_0(0,1)$. We know the interpolation space $$(X_0,X_1)_{1/2,2}=H^2(0,1).$$ I am wondering what is $$(X_0,X_2)_{1/2,2}=?$$ Would it be $H^2_0(0,...
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1answer
44 views

square weighted l^2

I am looking the sequence spaces $l^1$ and $$\{(x_k)_k: \|x\|_{sq}^2 := \sum_{k=1}^\infty k^2\cdot x_k^2 < \infty\}. $$ I am particularly interested in relations between their respective norms: It ...
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If two spheres are isometric, does there exist a bijective isometry $T:S\to S$ with $\|Tu-\alpha Tv\|_Y \leq \|u-\alpha v\|_X$ for all $\alpha>0?$

Let $$(S,\|\cdot\|) = \{(x,y)\in \mathbb{R}^2: \|(x,y)\| =1\},$$ that is, $S$ is the collection of all norm one vectors in $\mathbb{R}^2$ with respect to the norm $\|\cdot\|.$ Question: Let $\|\...
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185 views

Are biduals of spaces of differentiable functions on hypercubes Grothendieck?

Consider the space $E_n = C^1([0,1]^n)$ of continuously differentiable functions with the usual norm $$\max\{ \|f\|_\infty, \|f^\prime_{x_1}\|_\infty, \ldots, \|f^\prime_{x_n}\|_\infty\}.$$ making it ...
0
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1answer
192 views

Does this norm have a specific name? Banach space? References?

Let $(X,\mathscr{B},\mu)$ be a $\sigma$-finite measure space. Let $\gamma$ be a probability measure on $L_2(\mu)$ with $\mathrm{supp} \, \gamma = L_2(\mu)$ and existing first moment. Then $$ f \...
5
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1answer
157 views

Complemented subspaces constructed from finite pieces- part II

This is a follow up to: Complemented subspace constructed from finite pieces Suppose $Y=\overline{\cup E_n}$ is a closed subspace of a separable Banach space X, where each $E_n$ is a $n$-dimensional ...
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1answer
68 views

Complemented subspace constructed from finite pieces

Suppose $Y=\overline{\cup E_n}$ is a closed subspace of a Banach space, where each $E_n$ is a $n$-dimensional subspace, $K$-complemented in $X$, and for any $n$, $E_n\subseteq E_{n+1}$. Can one ...
4
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0answers
70 views

LlogL and Hardy space on the upper half plane

Set $\mathbb{T}$ the unit circle, $dm$ the Lebesgue measure on $\mathbb{T}$ and $\mathbb{C}^+=\left\{z\in \mathbb{C},s.t.\,\Im(z)>0 \right\}$ the upper half plane. It is well-known that the Cauchy ...
4
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2answers
241 views

Weak closure of subsets of the unitary sphere of a Banach space

Assume that $(X,\|\cdot\|)$ is a Banach space with $\|\cdot\|$ strictly convex. Define $S=\{x\in X:\|x\|=1\}$. Suppose that $\varepsilon>0$ and $x_0\in S$ and define $$ B_\varepsilon=\{x\in X:\|x-...
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2answers
366 views

If the closed unit ball of Banach space has at least one extreme point, must the Banach space the be a dual space?

Let $X$ be a Banach space. By Banach-Alaoglu and Krein-Milman Theorems, one can show that if $X$ is a dual space, then $X$ must have at least one extreme point of the closed unit ball. I am ...
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80 views

On functions obtained from Gaussian Quadrature integration

Fix an integer $n \ge 2$. Let $x_1,...,x_n$ s and $w_1,...,w_n$ s be the Gauss Quadrature nodes and weights respectively in the interval $[0,1]$ (https://en.wikipedia.org/wiki/Gaussian_quadrature) . ...
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1answer
130 views

“Compactness in Measure” in Function Spaces

In Chapter 4.9 of the book "Measures of Noncompactness and Condensing Operators" (Vol. 55 of Operator Theory: Advances and Applications), the authors mention the property "compactness in measure". ...
3
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3answers
132 views

Sum of subspaces is closed iff inclination is positive

It is a well-known result in functional analysis that the sum $M+N$ of two subspaces of a Banach space with $M\cap N=0$ is closed if and only if the inclination $$\widehat{(M,N)} := \inf_{x\in M, \|x\|...
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1answer
52 views

Reference on vector-valued convex conjugate

The following definition of convex conjugate is taken from Wiki: Let $X$ be a real topological vector space, and let $X^*$ be the dual space to $X.$ Denote the dual pairing by $$\langle \cdot ,...
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1answer
45 views

If the ranges of all f.d basis projections in $X$ are isometric to subspaces of $Y$, does it follow that $X$ is isometric to a subspace of $Y$?

Let $X$ be a Banach space with a Schauder basis, and $Y$ a Banach space. Let $P_N$ denote the coordinate projections relative to the basis of $X$, and let $X_N$ denote their ranges. Specifically, $\{...
4
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1answer
134 views

Interpolation of some Lebesgue spaces

When dealing with time-dependent PDEs, one often obtain that some quantity $E(t,x)$ belongs to a Lebesgue space $L^p_t(L^q_y)$, which means that $$\int_0^{+\infty}\|E(t,\cdot)\|_{L^q(\mathbb{R}^n)}^p ...
9
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1answer
267 views

Does the Banach space $( \ell ^2 \oplus \ell ^2 )$ have F.P.P?

The space $( \ell^2 ,\lVert \cdot \rVert _2 )$ is a Hilbert space. The space $X=(\ell^2 \oplus \ell^2 , \lVert \cdot \rVert_\infty )$ is a Banach space. Does X have fixed point property? (For any ...
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59 views

Separable measures on compact groups

Let us say that a (signed, finite) measure $\mu$ is separable if $L_1(|\mu|)$ is a separable Banach space. EDIT: Suppose that $G$ is a locally compact group such that each measure on $G$ is ...
1
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1answer
155 views

The continuum hypothesis and the diamond principle for $\aleph_1$

In [S. Shelah. Uncountable constructions for B.A. e.c. groups and Banach spaces. Israel J. Math. 51 (1985), 273-297], the existence of a special Banach space is proved, assuming the diamond principle ...
2
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0answers
106 views

Dual of the space of affine functions

Let $M^+(D)$ be the space of all positive measures on a closed convex subset $D$ of a locally convex topological vector space $E$. Two measure $\mu, \nu\in M^+(D)$ one can define a partial ordering $\...
4
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2answers
152 views

Compact images of nowhere dense closed convex sets in a Hilbert space

Let $B=-B$ be a nowhere dense bounded closed convex set in the Hilbert space $\ell_2$ such that the linear hull of $B$ is dense in $\ell_2$. Question. Is there a non-compact linear bounded operator ...
6
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3answers
296 views

Is there a version of Fischer-Riesz theorem for Banach space?

$( \Omega,F, P )$: a measurable space equipped with a finite measure $(B , \Vert \cdot \Vert) $ : a Banach space with $\mathcal{B}$ as its borelian $\sigma$-algebra $p$ : a constant bigger than $1$ ...
3
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1answer
119 views

Is it true that every Banach space has at least one extreme point that is normed by some point?

Definition: Let $X$ be a Banach space and $X^*$ be its continuous dual of $X,$ that is, $X^*$ contains all bounded linear functionals on $X.$ Denote $$B_{X^*} = \{x^*\in X^*: \|x^*\|_{X^*}\leq 1\}.$$...
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0answers
112 views

Complemented subspaces of $C(\beta\mathbb N\times \beta\mathbb N)$

Problem. Is there any complemented subspace in the Banach space $C(\beta\mathbb N\times\beta\mathbb N)$, not isomorphic to $c_0$, $c_0\oplus C(\beta\mathbb N)$, $C(\beta\mathbb N)$, $c_0(C(\beta\...
8
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0answers
121 views

A basis of the Banach space $L^p(\mathbb T^\omega)$ consisting of characters

Problem: For $1<p<\infty$, $p\ne 2$, has the complex Banach space $L^p(\mathbb T^\omega)$ got a Schauder basis consisting of characters of the compact topological group $\mathbb T^\omega$? (...
2
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1answer
153 views

Counter example about blow-up solution of DEs

Let $f(\cdot)$ be a continuously differentiable function over $\mathbb{R}$, and $u\in L^2_{loc}(0,\infty)$, $a\in \mathbb{R}$, and $x(t)$ solves the integral of $$\dot{x}(t)=ax(t)+f(x(t))+u(t), \quad ...
1
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2answers
175 views

About finite representability of Banach space

Can someone please tell me the brief sketch (or any known reference) of the following results? Why $\ell_2$ is finitely representable in any infinite-dimensional Banach space? Why every Banach space ...
3
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1answer
115 views

Is the norm of the Banach space projective tensor product of finite-dimensional C*-algebras a C*-norm?

Let $A$ and $B$ be two finite-dimensional C*-algebras. Let $\gamma$ denote the projective Banach space tensor product norm on the algebraic tensor product $A\odot B$, so $\gamma(t)=\inf\{\sum_{i}\|...
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2answers
458 views

Multiple of identity plus compact

Is there an example of a bounded operator $T\in\mathcal{B}(H)$, where $H$ is a separable complex Hilbert space, such that no restriction to an infinite dimensional closed subspace is multiple of ...
6
votes
1answer
234 views

Convolution with semigroup: does this belong to the Sobolev space $W^{1,1}$?

Let $X$ be a Banach space, $T(t)$ be a strongly continuous semigroup on $X$, and $f\in L^1(0,\tau;X)$. It has been implied that the integral $$v(t)=\int_0^t T(t-s)f(s)ds,\quad t\in [0,\tau]$$ is not ...
1
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1answer
90 views

Optimal estimate in trace norm

Let $x,y$ be vectors of some Hilbert space of unit length. Then we can consider the projection $P_x:=\langle \bullet, x \rangle x$ and similarly $P_y.$ Assume then that we know that $\left\lVert x-...
3
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0answers
90 views

Fixed-point properties for affine actions of topological groups

T. Mitchell [Illinois J. Math. 14 (1970) 630--641] defined four properties of a topological semigroup, and in particular of a topological group $G$. Two of them are: (F2) Every jointly continuous ...
10
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1answer
263 views

Criterion for a Banach algebra to be finite dimensional

Let $A$ be a Banach algebra (say, complex and unital) and suppose that every (closed) commutative subalgebra of $A$ is finite dimensional. Question. Does it follow that $A$ is finite dimensional? ...
0
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2answers
297 views

Does point-wise weak convergence give weak convergence in $L^2(I;X)$?

Let $X$ be a separable reflexive Banach space, $F$ be a locally Lipschitz nonlinear operator on $X$ that is weakly continuous on $X$, and $u_n$ are $u_n$ weakly converges to $u$ on $L^2(0,T;X)$. Now, ...
1
vote
1answer
84 views

Orthogonal complement vector space

Let $X$ be a vector space contained in $H^{1}(\mathbb R^d),$ then we can study $X^{\perp_{L^2}}:=\left\{ \xi \in L^2; \langle \xi, x \rangle_{L^2} =0 \ \forall x \in X \right\}$ and $X^{\perp_{H^{-...
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0answers
188 views

Which subspaces of $\ell_p^n$ are isometric?

This question is similar to the one asked here: Extending linear isometries from subspaces of $\ell_p^n$ Let $p$ be an even integer. Let $X,Y$ be subspaces of $\ell_p^n$, and let $U : X \to Y$ be a ...
17
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2answers
423 views

On a special type of normed linear spaces

Let $(V,\|.\|)$ be a normed linear space such that for every group $(G,*)$, every function $f:G \to V$ satisfying $$ \|f(x*y)\|\ge \|f(x)+f(y)\|,\qquad\forall x,y\in G,\tag{Z} $$ is a group ...
4
votes
1answer
109 views

Closure of tensor product /tensor product semigroup

In this reference the following claim is made in Remark 2 Let $A,B$ be closable operators on Banach spaces $X,Y$, then $A \otimes 1$ and $1 \otimes B$ are closable operators on the Banach space $X \...
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0answers
60 views

Nonlinear maps in Riesz Thorin theorem

The Riesz Thorin theorem allows us to interpolate between $L^p$ spaces and the usual assumption is that the map $T$ is linear. What I was wondering about is whether this is because otherwise you do ...
0
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1answer
87 views

Why ker(1) is a semi M-ideal in $\ell_1$?

A subspace $Y$ of $X$ is said to be semi M-ideal if $\exists$ a projection $P$ (not necessarily linear) from $X^*$ to $Y^\perp$ such that $\|x^*\|=\|Px^*\|+\|x^*-Px^*\|$. And also, $P(\lambda x^*+Py^*)...
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2answers
562 views

If $E\oplus_\phi E \cong E\oplus_\psi E,$ does it imply that $\phi= \psi$?

Let $E\neq \{0\}$ be a Banach space. For each $p\in[1,\infty), $ we define $$E\oplus_p E = \{(x,y): x\in E, y\in E, \|(x,y)\| = \sqrt[p]{\|x\|^p + \|y\|^p}\}.$$ Let $F$ be another Banach space. By $E\...
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1answer
99 views

Does there exists an extreme point $(a_1^*,…,a_n^*)$ of $B_{\mu^*}$ such that $a_i^*\neq 0$ for all $1\leq i\leq n$ and $\sum_{I=1}^n a_i^*a_i=1?$

Fix a natural number $n\geq 1.$ Let $\mu$ be a norm on $\mathbb{R}^n$ satisfying $$\mu(0,...,0,\stackrel{i}{1},0,...,0) = 1 \quad\text{for all }1\leq i\leq n.$$ Let $$B_{\mu} = \{(a_1,...,a_n)\in \...
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1answer
105 views

Do functions exist and are they dense? Or does it depend on the basis?

Consider an orthonormal basis $(\varphi_n)_{n \in \mathbb N}$ of $L^2(\mathbb R).$ We consider the functionals $\Phi_n$ given by $$ C^b(\mathbb R) \ni f \mapsto \left\langle \varphi_n, f \varphi_{n+1}...
3
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0answers
42 views

The dual relationships between strictly singular operators and strictly cosingular operators

Recall that an operator $T:X\rightarrow Y$ between Banach spaces is called strictly singular if if it is not an isomorphism when restricted to any infinite-dimensional closed subspace of $X$. An ...
4
votes
1answer
146 views

Approximation property counterexamples? (Also: relation to tensor products)

I remember reading somewhere (but unfortunately, I've forgotten where it was) that the canonical map from the (completed) projective tensor product of two Banach spaces to the (completed) injective ...
2
votes
1answer
100 views

A formula for vector valued measurable functions

Let $B_{\infty}(\Omega)$ be the space of bounded measurable functions on the measurable space $\Omega$. For a given Banach space $X$, let us denote $B_{\infty}(\Omega,X)$ by the set of all bounded ...
1
vote
1answer
105 views

Compactness of operators and norming sets

Originally asked on MSE. Let $T$ be a linear map from a normed space $E$ into a Banach space $F$. Let $D\subset \overline{B}_{F^{\ast}}$ be norming, i.e., there is $r>0$ such that $\sup\limits_{v\...
1
vote
1answer
46 views

Infinitely many independent functions that are only frequency localized?

A function $f \in L^2(\mathbb R^d)$ will be called $K$-frequency localized if the following inequality holds $$\int_{\mathbb R^d} \lvert \widehat{f}(x) \rvert^2 x^2 \ dx \le K \int_{\mathbb R^d} \...
0
votes
1answer
149 views

Central limit theorem in Banach space in scheme of series

I wonder whether Theorem 2 from the paper J. Zinn, Annals of Probability, 1977, vol. 5, 283-286 can be extended to the CLT for a scheme of series. (The paper is available in the web.) Let $G$ be ...
3
votes
1answer
149 views

Does Bishop-Phelps Theorem hold for extreme points (slightly different version)?

Recall the Bishop-Phelps Theorem. Bishop-Phelps Theorem: Let $B\subseteq E$ be a bounded, closed, convex subset of a real Banach space $E.$ Then the set $$\{e^*\in E^*: e^* \text{ attains its ...