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.
1,592
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Finite-dimensional subspaces of $c_{0}$
Let $M$ be a finite-dimensional subspace of $c_{0}$, and let $\varepsilon>0$.
Question. Does there exist a finite rank projection from $c_{0}$, of norm $\leq 1+\varepsilon$, onto a subspace $N$ of ...
3
votes
1
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392
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Uniform smoothness and twice-differentiability of norms
To get to the simplest case, consider a norm $\|\cdot\|$ over $R^n$ that is uniformly smooth of power-type 2, that is, there is a constant $C$ such that $$\frac{\|x+y\| + \|x - y\|}{2} \le 1 + C \|y\|^...
10
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Extremal bases in finite-dimensional Banach spaces
Definition. A basis $e_1,\dots,e_n$ for a Banach space $X$ is called extremal if there exists a point $s$ in the unit sphere $S_X=\{x\in X:\|x\|=1\}$ such that for every $i\in\{1,\dots,n\}$ the ...
6
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Mazur-Ulam bases in finite-dimensional Banach spaces
Definition. A basis $e_1,\dots,e_n$ of a finite-dimensional Banach space $X$ is called Mazur-Ulam if all vectors $e_1,\dots,e_n$ have norm one and every self-isometry $f:S_X\to S_X$ of the unit sphere ...
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1
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127
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Both $\ell_\infty$ and $L_\infty$ belong to $\mathcal{B}_1$ [duplicate]
I am looking for a resource or some hints on why the two normed spaces $\ell_\infty$ and $L_\infty$ belong to the family $\mathcal{B}_1$, that is,
they are of the family of Banach spaces $X$ such that ...
1
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0
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66
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Inequality of exponentials of Banach operators
(I have moved this question from Stackexchange).
Given the operators $\{A_j\}$ in a Banach algebra and a positive integer $p$, let
\begin{equation}
g=\exp\left(\frac{1}{n}\sum_{j=1}^p A_j\right)\quad\...
2
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1
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132
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How to characterize the order convergence in Bochner-integrable functions space?
Let $(\Omega,\Sigma,\mu)$ a finite measure space. We want to characterize the order convergence (for sequences) in Bochner integrable functions space $L^1(\mu,X)$, $X$ Banach lattice.
In $L^p$ we have:...
5
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1
answer
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Embedding of a Banach space into a Hilbert space
Let $\mathbb H$ be a Hilbert space and let $\mathbb B$ be a Banach space continuously embedded in $\mathbb H$ and distinct from $\mathbb H$. Is it true in general that $\mathbb B$ is an $F_\sigma$ of ...
3
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1
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Banach Mazur distance between the cube and the cross-polytope in the dimensions for which a Hadamard matrix exists
The Banach-Mazur distance between two centrally symmetric convex bodies $K,L\in\mathbb{R}^n$ can be defined as
$$ d(K,L) = \inf \{ r : \exists T\colon \mathbb{R}^n \to \mathbb{R}^n \text{ linear such ...
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Is this Beppo-Levi curl space a Banach space?
Let us define the quotient space:
$$ V = \{ \mathbf{u} \in L^2_{loc}(\mathbb{R}^3; \mathbb{R}^3) : \operatorname{curl} \mathbf u \in L^2(\mathbb{R}^3; \mathbb{R}^3) \} / \nabla H^1_{loc}(\mathbb{R}^3)....
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120
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Criteria for $(V \otimes W)^* \cong V^* \otimes W^*$ in Banach spaces
Let $V$ and $W$ be Banach spaces.
$V^* \otimes W^*$ embeds into $(V \otimes W)^*$ (projective tensor product). I am looking for criteria for it to be an isomorphism.
If $V$ and $W$ are $C^*$-algebras, ...
1
vote
1
answer
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What's the size of non standard monad for weak topology?
There have been several works characterizing weak topology by nonstandard analysis, which give rise to the following monad ($X$ is a Hilbert space):
$$\mu(0) = \{y\in{}^{*}X: \forall x\in X ~~ \...
3
votes
1
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The weak*-convergence of the summing basis of $c_{0}$
Suppose that $(x_{n})_{n}$ is a sequence in a Banach space $X$. We let $\textrm{clust}_{X^{**}}((x_{n})_{n})$ be collection of all the weak*-limit points of $(x_{n})_{n}$ in $X^{**}$.
Let $(e_{n})_{n}$...
2
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1
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Is the union of good equivalence relations on a compact space good?
Let $X$, $Y_1$ and $Y_2$ be a compact Hausdorff spaces and let $\varphi_i:X\to Y_i$ be a continuous surjection (and so a quotient map).
Let $\sim$ be the minimal closed equivalence relation on $X$ ...
10
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1
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A quantity measuring the separability of Banach spaces
Let $X$ be a Banach space. It is natural for us to introduce a quantity measuring the separability of sets as follows: for a subset $A$ of $X$, we set
$\textrm{sep}(A)=\inf\{\epsilon>0: A\subseteq ...
2
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3
answers
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How do I apply Brouwer fixed-point theorem in this claim?
Let $\zeta:\mathbb{R}\to [0,+\infty)$ be a continuous non-negative function such that $\zeta(0)=0$ and $\tau\mapsto \zeta(\tau)\tau$ is a non-decreasing differentiable function whose derivative is ...
30
votes
1
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Functional-analytic proof of the existence of non-symmetric random variables with vanishing odd moments
It is known that a random variable $X$ which is symmetric about $0$ (i.e $X$ and $-X$ have the same distribution) must have all its odd moments (when they exist!) equal to zero. The converse is a ...
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1
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When is a complex of Banach spaces exact as condensed abelian groups?
Lectures on Condensed Mathematics, Theorem 3.3 says that for any compact Hausdorff space $S$, the cohomology $H_{\mathrm{cond}}^i(S,\mathbb R)=0$ for $i>0$ and $H_{\mathrm{cond}}^0(S,\mathbb R)=C(S,...
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1
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143
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$l_{1}$-block basic sequences in Banach spaces with an unconditional basis
Let $X$ be a Banach space with an unconditional basis $(x_{n})_{n}$.
Question. If $X$ contains a subspace isomorphic to $l_{1}$, does $(x_{n})_{n}$ admit a block basic sequence equivalent to the unit ...
3
votes
1
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436
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Fréchet vs. Carathéodory differentiability on Banach spaces
It is well-known that in the finite-dimensional case one can use the notion of Fréchet differentiability and Carathéodory differentiability interchangeably. See for example the 194 AMM article Frechet ...
3
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0
answers
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Regularity of the dependence of the flow on the vector field definining it
Let $M$ be a smooth compact manifold and $k \geqslant 1$.
Define $\mathfrak{X}^k(M,TM)$ to be the set of vector fields $M \rightarrow TM$ of class $C^k$. As $M$ is compact, endowing $\mathfrak{X}^k(M,...
13
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3
answers
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Disintegrations are measurable measures - when are they continuous?
This is a sequel to another question I have asked.
The notion of disintegration is a refinement of conditional probability to spaces which have more structure than abstract probability spaces; ...
0
votes
0
answers
154
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Sequence of functions tending to zero in L^2
Let us consider a sequence of functions $f_n : (0,1)\times (0,1) \to \mathbb{R}$ in $L^2((0,1)\times (0,1))$ satisfying the following condition:
$$
\lim_{n \rightarrow \infty}\int_{1/j}^{1 - 1/j}\...
2
votes
0
answers
60
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Decomposition of the Orlicz norm into sequential norm
I am bearing seeking for a sequential decomposition of the norm in Orlicz space.
Let me state what is known in the particular case of Lebesgue space $L^p(\Bbb R^d)$.
Given $u\in L^p(\Bbb R^d)$ let
$$n\...
2
votes
1
answer
107
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Control on dimension of image
Let $f:E\rightarrow F$ be a map between Banach spaces E and F; E finite dimensional (>0) and F infinite dimensional. Let $F$ be equipped with its weak topology and suppose that $f$ is strong-weak ...
0
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1
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780
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Specific criterion for the sum of two closed sets to be closed
Let $Y$ and $Z$ be two closed subspaces of a Banach space $X$ with $Y\cap Z=\{0\}$.
I know that $Y+Z$ is a closed subspace of $X$ $\iff \exists \alpha > 0:\quad \lVert y\rVert \le \alpha\lVert y+z\...
5
votes
2
answers
242
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Is there a topology that makes every basic sequence null?
Let $E$ be a Banach space. Let $F$ be the collection of all $f\in E^*$ such that $\left<f,e_n\right>\to 0$, for every normalized basic sequence $\{e_n\}$. It is easy to see that $F$ is a closed ...
1
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1
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Is a topology sandwiched between two norms compactly generated?
Recall that a Hausdorf topological space $X$ is called compactly generated if any set whose intersections with compacts are compact is closed. Locally compact and first countable spaces are compactly ...
5
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2
answers
295
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A quantity measuring the reflexivity of Banach spaces
Recall that $(y_{n})_{n}$ is a convex block subsequence of a sequence $(x_{n})_{n}$ in a Banach space $X$ provided that there exists a strictly increasing sequence of positive integers $(k_{n})_{n}$ ...
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When is a Banach space a Hilbert space?
Let $\mathcal{X}$ be a real or complex Banach space.
It is a well known fact that $\mathcal{X}$ is a Hilbert space (i.e. the norm comes from an inner product) if the parallelogram identity holds.
...
5
votes
1
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If a subspace $F$ is contained in a subspace $G$, and $H$ is close to $G$, can we choose a subspace of $H$ close to $F$?
Let $E$ be a Banach space. Recall that the collection of all closed linear subspaces of $E$ can be turned into a metric space in a number of ways. In particular, consider the notion of a gap: if $G$ ...
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Is a Banach lattice isomorphic to a Hilbert space in fact a Hilbert lattice?
The title says it all:
Let $E$ be a Banach lattice, which is isomorphic to a Hilbert space (as normed spaces). Is there an equivalent Hilbert norm on $E$, which still makes it a Banach lattice with ...
2
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0
answers
187
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What is the smallest number of nowhere dense affine subsets covering a topological group?
$\DeclareMathOperator\cov{cov}\newcommand\A{\text A}$A subset $A$ of a group $G$ is called affine if $A=xHy$ for some subgroup $H\subseteq G$ and some $x,y\in G$.
Given a non-discrete topological ...
3
votes
2
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393
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$C[0,1]$ is not a Grothendieck space
A Banach space $X$ is called a Grothendieck space if $\text{weak}^{*}$-null sequences in $X^{*}$ are weakly null. Some of the classical Grothendieck spaces are the $C(\Omega)$ spaces if $\Omega$ is ...
0
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0
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Convergence of characteristic functions vs. weak convergence of measures and the Ito-Nisio theorem
In section 2.6 of Linde's Probability in Banach Spaces: Stable and Infinitely Divisible Distributions the author is pointing out that in infinite-dimensional Banach spaces the convergence of ...
0
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1
answer
131
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Lower semi-continuity of induced function on sequences
Let $f:X\rightarrow [0,\infty)$ be (resp. weakly) lower semi-continuous on the reflexive Banach space $X$. Let $\ell^p(X)$ denote the space of $p$-summable sequences in $X$, i.e.: $\sum_{n=1}^{\infty}...
12
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Subtracting the weak limit reduces the norm in the limit
Question
Let $X$ be some reflexive Banach space. Suppose $x_n$ is some sequence in $X$ that weak converges to some $y \neq 0$. Is it the case that
$$ \limsup \|x_n - y\| < \limsup \|x_n\| ?$$
...
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2
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Weak convergence in the intersection of Lebesgue spaces or Sobolev spaces
Let $B:=B_1\cap B_2\cap...\cap B_n$, where each $B_j$ is a reflexive Lebesgue space or Sobolev space (such as $L^4$, $H^1$, etc.) on a domain in $\mathbb{R}^d$. Then $B$ is a Banach space endowed with ...
0
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0
answers
44
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Measuring non-Cauchyness of sequences on $c_{0}$
Let $(x_{n})_{n}$ be a sequence in the closed unit ball $B_{c_{0}}$ of $c_{0}$. Let $\epsilon>0$. My question: Is there a subsequence $(z_{n})_{n}$ of $(x_{n})_{n}$ so that $\textrm{ca}((z_{n})_{n})...
2
votes
1
answer
122
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An inequality of spacetime Banach space for non-linear Schrodinger equation
I am reading a proof for the existence of a solution to the Local Cauchy problem of the non-linear Schrodinger equation
$$
i\partial_t u+\Delta u +\epsilon u |u|^{2} = 0 \\
u(x,0)=u_0(x)
$$
The ...
1
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0
answers
63
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Embedding a normed space as a hyperplane
Let $X$ be a real normed space and suppose that $X$ is a closed hyperplane of a bigger space $\tilde X$. Given any unit vector $u$ in $\tilde X\setminus X$, consider the function $p:X\to\mathbb R$ ...
1
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0
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Generalizations of the Wiener Tauberian Theorem to Musielak-Orlicz spaces
Musielak-Orlicz spaces provide a generalization of the usual $L^p$ spaces on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ to spaces of functions for which the Luxemburg norm
$$
\|f\|_M:=\inf\left\{\lambda &...
0
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0
answers
68
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Convex optimization under asymmetric loss in infinite dimensional space
The following problem is common in financial economics
$$ \min_{m \in L^2} \mathbb{E}[ \phi(y(\theta)-m)] \quad \text{s.t. } \mathbb{E}[ mx ]= q $$
That is, given a random variable $y(\theta)$ ($\...
3
votes
0
answers
144
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Non uniqueness of center of the Banach-Mazur compactum
In "The Banach-Mazur distance to the cube and the Dvoretzky-Rogers factorization" Szarek and Bourgain prove a proportional Dvoretzky-Rogers factorization :
Given $1>\delta>0$ , there ...
0
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0
answers
103
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A noncontinous algebra map between Banach algebras
What is an example of two Banach algebras $A$ and $B$, and an algebra map $\phi:A \to B$ which is not continuous?
1
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1
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163
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Asymptotic models and passing to sub-arrays
If, for all $k\in\mathbb{N}$, $(x_{i}^{k})_{i=1}^{\infty}\in X^{\mathbb{N}}$ is normalized and $M$-basic and if, in addition, for all $k\leq i_{1}<i_{2}<\ldots$ the diagonal sequence $(x_{i_{k} }...
2
votes
1
answer
171
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Does set of finitely additive probability measures embed linearly into a strictly convex dual Banach space?
I am trying to better understand a condition that appears in Theorem 1 of this paper.
Let $K$ be a convex and compact subset of a locally convex tvs. The condition is:
$K$ embeds linearly into a ...
0
votes
1
answer
89
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A characterization of Grothendieck spaces via convex block subsequences
Recall that $(y_{n})_{n}$ is a convex block subsequence of a sequence $(x_{n})_{n}$ in a Banach space $X$ provided that there exists a strictly increasing sequence of positive integers $(k_{n})_{n}$ ...
9
votes
1
answer
305
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Closedness of linear image of positive L1 functions
Let $\mathcal X$ be the Banach space of $L^1$ functions on some probability space, $\mathcal Y$ be some other Banach space, $T:\mathcal X\to \mathcal Y$ be some surjective continuous linear map, $\...
0
votes
1
answer
181
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Integral estimate (inequality) with a Schwartz function
$\DeclareMathOperator\supp{supp}\newcommand\abs[1]{\lvert#1\rvert}\newcommand\Bigabs[1]{\Bigl\lvert#1\Bigr\rvert}$Given a Schwartz function $f \in \mathcal{S}(\mathbb{R})$ with $\supp(f) \subseteq [-A,...