# 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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### Boundary points in $\overline{\operatorname{conv}\{z_i\}_{i\in I}}$

Let $X$ be an infinitely-dimensional Banach space and $\{z_i\}_{i\in I}$ be a set of linearly independent points in $X_{\leq 1}$, the closed unit ball of $X$. $I$ the index set is not necessarily ...

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### Weak convergence in $H^{1}$ implies different convergence in $L^{p}$?

Suppose I have a sequence $\{f_{n}\}_{n\in \mathbb{N}} \subset H^{1}(\mathbb{R}^{d})$ which converges weakly to $f$ in $H^{1}(\mathbb{R}^{d})$, in the sense that $\langle f_{n},\varphi \rangle_{L^{2}}+...

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### Banach-Mazur distances between quantum tori

Let $u,v\in B(L_2(\mathbb T))$ defined as $u(f)(z)=zf(z)$ and $v(f)(z)=f(ze^{-2\pi i\theta})$ for $z\in\mathbb T$ where $\theta\in\mathbb R\setminus\mathbb{Q}$. Denote the $C^*$-algebra generated by $...

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### Continuity of linear map on tensor product spaces with different norm properties

I originally asked this question on StackExchange, but I think that it may be more suitable to here.
Let $V$ and $U$ be Banach spaces. I'm considering a linear map $\phi: V \rightarrow U$, and ...

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### When is $W^{1,p}(\Omega)$ a Banach algebra?

Let $\Omega \subseteq \mathbb{R}^d$ be bounded and open with $\partial\Omega$ Lipschitz, $1<p<\infty$.
My question: knowing that $f,g\in W^{1,p}(\Omega)$ for what $p$ can we conclude that $f\...

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### Pointwise convergence and disjoint sequences in $C(K)$

Let $K$ be a Hausdorff compact space and let $C(K)$ be the space of continuous real-valued functions on $K$. A sequence $(h_n)$ in $C(K)$ is called almost disjoint if there is a sequence $(g_n)$ with ...

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### Banach tori: classification up to Fréchet homeomorphisms

Consider the set $T$ in $l_p$ defined as closure of
\begin{equation}
T = \{ (x_1,\dotsc,x_n,\dotsc): x_j = \frac{1}{2^{(j/p)}} e^{it_j}, t_j \in \mathbb{R}/\mathbb{Z} \}.
\end{equation}
This seems to ...

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### Analogy between quasi-injective modules & extensible Banach spaces

Let $X$ be a module. $X$ is said to be quasi-injective if every homomorphism $h:A\to X$ from any submodule $A\subseteq X$ has an extension to an endomorphism $\tilde{h}:X\to X$.
A module $X$ is quasi-...

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### Convexity of an equivalent norm

Let $X=l_2$ with usual norm $\|\cdot\|_2$. We define a subspace of $X$ as $D=conv (B_{l_2} \cup B),$ where $B = \{ (x_n) \in l_2 : \sum_{n=1}^\infty \frac{n}{2} x_n^2 \leq 1\}$, conv is the convex ...

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### Zero flux along lines

I am considering the $L^1$ ball in $\mathbb{R}^d$, and a conservative vector field $V$ on it, which arises as the gradient of a bounded, almost-everywhere Lipchitz-function. Denoting by $e_i$ as the i’...

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### Distance between convex hulls in a bounded closed convex set

Let $X$ be an infinite-dimensional Banach space and $C\subseteq X$ be a bounded closed convex subset. Let $\{z_i\}_{i\in\mathbb{N}}$ be a sequence of linearly independent points in $C$ and for each $n\...

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### Infinite dimensional homology theory for submanifolds of Hilbert and Banach spaces

Is there a version of homology theory for spaces for which explicitly infinite dimensional "cells" are allowed?
The spaces in question include e.g.
\begin{equation}
X = (x: x \in l_2: p_i(x) ...

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### Is $L^p_\text{loc} (Y)$ dense in $(L^0(Y), \hat \rho)$?

Below we use Bochner measurability and Bochner integral. Let
$(Y, d)$ be a separable metric space,
$\mathcal B$ Borel $\sigma$-algebra of $Y$,
$\nu$ a $\sigma$-finite Borel measure on $Y$,
$(Y, \...

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### Is there $r>0$ such that the norm $[f]_r:=\sup_{n \ge 1} \|1_{B(y_n, r)} f\|_{L^p}$ is equivalent to $\|f \|:=\sup_{y \in Y}\|1_{B(y,1)}f\|_{L^p}$?

Below we use Bochner measurability and Bochner integral. Let
$(Y, d)$ be a separable metric space,
$\mathcal B$ Borel $\sigma$-algebra of $Y$,
$\nu$ a $\sigma$-finite Borel measure on $Y$,
$(E, |\...

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### Is the embedding $i: (L^p_\text{loc} (Y), \| \cdot \|_{L^p_\text{loc}}) \to (L^0(Y), \hat \rho)$ continuous or Borel measurable?

Below we use Bochner measurability and Bochner integral. Let
$(Y, d)$ be a separable metric space,
$\mathcal B$ Borel $\sigma$-algebra of $Y$,
$\nu$ a $\sigma$-finite Borel measure on $Y$,
$(Y, \...

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### The set of measurable functions together with convergence in measure is a completely metrizable abelian topological group

Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete $\sigma$-finite measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple ...

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### Assume $f(x, \cdot) \in L^p_{\text{loc}} (Y)$ for a.e. $x \in X$. Is $F: X \to L^p_{\text{loc}} (Y), x \mapsto f(x, \cdot)$ Bochner measurable?

Below we use Bochner measurability and Bochner integral. Let $T>0$ and $p \in [1, \infty)$. Let $X :=[0, T]$ and $Y:= \mathbb R^d$. Let $L^p_{\text{loc}} (Y)$ be the space of measurable functions $...

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### Is there a version of dominated convergence theorem for local $L^p$ spaces?

Fix $p \in [1, \infty)$. Let $(L^p (\mathbb R^d), \|\cdot\|_{L^p})$ be the Lesbesgue space of $p$-integrable real-valued functions on $\mathbb R^d$. Let $\tilde L^p (\mathbb R^d)$ be the space of ...

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### Understanding $k$-rotundity

We know that the definitions of $k$-Uniform rotundity ($k$-UR) or locally $k$- uniform rotundity (L$k$-UR) act to find the volume in higher k-dimensional spaces by defining $V(x_1, ..., x_{k+1})$ (For ...

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### Injective envelopes of 1-extensible spaces

Please read this post as a naive follow up on a previous question.
Let $X$ be a Banach space and let $(I(X),\alpha)$ denote its injective envelope (e.g., CohenLacey1969). A low hanging fruit is the ...

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### For $\mu$-a.e. $x \in X$, the sequence $(f_n(x, \cdot))_n$ is Cauchy in $L^1 (Y)$. Then $(f_n)$ is Cauchy in $L^1 (X \times Y)$

Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the ...

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### A notion of restricted injectivity for Banach spaces

I apologize in advance if this is well-known.
Let $X$ be a Banach space. Let's call only for this post that $X$ is self-injective if for every closed subspaces
\begin{equation}
A\subseteq B\subseteq X ...

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### Property (H) in the dual norm

Consider the Hilbert space $l_2$ with an equivalent norm
$$\Vert x \Vert = \max \{2 \Vert x \Vert_1, \Vert x \Vert_2 \},$$
where $\Vert x \Vert_1 =( \sum_{n=2}^\infty x_n^2 )^{\frac{1}{2}}$ and $\Vert ...

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### Existence for a nonlinear evolution equation with a monotone operator that is not maximal

We consider the nonlinear evolution equation
$$
\dot{u}(t) + Bu(t) = 0, \quad u(0)=0
$$
with
$$
A: \mathcal{C}(\Omega)\to \mathcal{M}(\Omega),\; p \mapsto \arg\min_{\mu\in\partial\chi_{\{||\...

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### Renorming on a separable Banach space

Let us consider the sequence space $c_0$ with the equivalent norm
$$\Vert x \Vert^2 = \max_{i\ge1} \vert x^i \vert^2 + \sum_{i=2}^{\infty} 2^{-i+1} \vert x^i \vert^2 $$
for $x=(x^1,x^2,\ldots)\in c_0$....

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### Examples of topologically non trivial complete submanifolds in infinite dimensional Banach spaces

In infinite dimensional Banach spaces, many analogies of classical sets are topologically trivial ( even contractible). E.g., infinite dimensional spheres are contractible by Y. Benyamini, Y. ...

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### The real and the imaginary part of a vector

In an infinite-dimensional Banah space $(X, \|\cdot\|)$ with a countable Schauder basis $\{x_n\}$, define:
$$
F_r: \operatorname{Span}(\{x_n\}) \rightarrow \operatorname{Span}(\{x_n\}), \hspace{0.3cm} ...

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### Infimum of norms of elements in a hyperplane

In a Banach space X, given a norm one bounded linear functional $f$ and $c\in \mathbb{C}\backslash \{0\}$, define $H = \big\{ x\in X \,\vert\, f(x) = c\big\}$ and $\inf H$ = $\inf_{h\in H} \|h\|$.
Is ...

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### Does every Banach space admit a continuous (not necessarily equivalent) strictly convex norm?

Trying to find and answer to this question, I have encountered two more-studied problems.
The first is to find when a Banach space admits an equivalent uniformly convex norm. The answer is that for ...

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### Are there any books or literature on norms over measure space?

Consider the space of signed measures over some abstract space, we know the total variation norm makes the space Banach (I guess). So are some other norms. Are there some books or literature studying ...

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### Does this variant coincide with the usual Hölder space?

$\newcommand{\NN}{\mathbb N} \newcommand{\RR}{\mathbb R}$
Let $\alpha \in (0, 1]$ and $d, j \in \NN^*$.
The usual Hölder space $C^{j, \alpha} := C^{j,\alpha} (\RR^d; \RR)$ is defined as the space of ...

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### Is this generalization of Fredholm alternative true?

Let $E,F$ be Banach spaces. Let $\mathcal L(E, F)$ be the space of bounded linear operators from $E$ to $F$, and $\mathcal K(E, F)$ its subspace consisting of compact operators. For a linear map $T$, ...

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### A counterexample showing $BV_p \neq AC_p$

I am trying to work through a supposedly simple counterexample given in papers by Love and Gehring regarding a $p$-power generalization of bounded variation and absolute continuity.
Let $p > 1$. ...

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### Extremal problem for 2-dimensional lattices

Given a lattice $L$ in a Banach space $(B,\|\;\|)$, one denotes by $\lambda_1(L)$ the least norm of a nonzero element in $L$, and by $\lambda_k$ the least $\lambda$ such that there is a linearly ...

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### LF or LB space that happens to be finite dimensional

Let $\{V_n\}_{n=1}^\infty$ be a collection of finite dimensional vector subspaces of $L^2[0,1]$ such that $V_n \subset V_{n+1}$ and $\bigcup_{n=1}^\infty V_n$ is dense in $L^2[0,1]$. Suppose further ...

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### Is $B \cap L^2\big( (0,T) \times (0,1)\big)$ closed in $L^2\big( (0,T) \times (0,1)\big)$?

I need help proving that the set $B \cap L^2\big( (0,T) \times (0,1)\big)$ is a closed subset of $L^2\big( (0,T) \times (0,1)\big)$, where $B$ is defined as:
$$B=\Big\{x \in L^{\infty}\big(0,T;L^1(0,1)...

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### Examples of non $w^{*}$-closed complemented subspaces of a dual Banach space that are also dual spaces

Let $Y$ be a complemented, but not $w^{*}$-closed, subspace of a Banach space $X$. It is known that certain such $Y$ are not dual spaces.
Question: What are interesting examples of subspaces of the ...

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### Weakly compact operators into $c_0$ and other separable spaces

A Banach space $E$ is called Grothendieck if every weak* convergent sequence in the dual space $E^*$ is weakly convergent. A typical example of a Grothendieck space is $\ell_\infty$. Diestel proved ...

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### Weak*-separability of the unit ball of $X’$ and density characters and cardinalities of $X$ and $X’$

(This question has also been asked on Math StackExchange.)
Let $X$ be a Banach space, $X’$ be its continuous dual such that its unit ball is weak*-separable. I’ve been wondering what can be said about ...

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### Is there a bounded sequence $(e_n)$ such that $e_n \in E_n$ and that $(e_n)$ does not have any convergent subsequence?

Let $(E, |\cdot|)$ be an infinite-dimensional Banach space. Assume that
$T:E\to E$ is a compact (bounded linear) operator, and
$(\lambda_n)$ is a sequence of distinct eigenvalues of $T$.
Let $E_n$ ...

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### Complemented subspaces of a dual Banach space

Let $\kappa$ be an infinite cardinal number and by $\mathcal{B}(\kappa)$ denote the class of all Banach spaces of density $\kappa$.
My question reads as follows:
Does there exist $\kappa$ for which ...

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### Is $I-S$ in my attempt of Fredholm alternative injective?

Let $E$ be a Banach space. Let $\mathcal K(E)$ be the space of all compact (bounded linear) operators from $E$ to $E$. For a linear map $T$, we denote by $R(T)$ its range and by $N(T)$ its kernel. Let ...

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### Is $\Lambda:= \pi_2 \circ \pi_1:E \to L$ surjective?

Let $E$ be a Banach space. For a linear map $T$, we denote by $R(T)$ its range and by $N(T)$ its kernel. Let $I:E \to E$ be the identity map. Let $T:E \to E$ be a compact (bounded linear) operator. ...

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### Norm distance in a Banach space

Consider the Hilbert space $l_2(\mathbb{N})$ under the square summable norm $\Vert \cdot \Vert_2.$ Let us define a new norm $||| \cdot ||| $ equivalent to $\Vert \cdot \Vert_2$ such that the closed ...

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### Finite dimensionality of a subspace

Let $c>0$ and let $Y$ be the space of all distributions of compact support in $(-1,1)$ with singular support at $\{0\}$. Let $X$ be subspace of $Y$ such that for any $\phi \in X$ there holds:
$$ \...

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### Compact embedding of homogeneous weighted Sobolev spaces

Let $n\geq 2$ and let $\Omega$ be the open unit ball with the origin removed. For each $\delta>0$ and each $u\in C^{\infty}(\Omega)$ let us define
$$ \|u\|^2_{L'^1_\delta(\Omega)}= \int_{\Omega} |x|...

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### Ultraproducts of Banach spaces versus model theoretic ultraproduct

Reading about ultraproducts in model theory and in Banach spaces leads to two distinct definitions. E.g., for an ultrapower given by an ultrafilter $\mu$ on $\mathbb{N}$, both notions of ultrapower ...

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### An example of a non rigid Banach algebra

A Banach algebra $A$ is called a rigid Banach algebra if for every injective Banach algebra morphism $J:A\to A$ we have either $\overline{J(A)}$ is ismorphic to $A$ or it does not contain ...

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### Does there exists a smooth reflexive infinite dimensional Banach space that is not strictly convex

It is known that in a reflexive Banach space, if the norm is strictly convex, then its dual will be smooth Banach space, and if the norm is smooth, then the dual norm is strictly convex.
We can find ...

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### Smoothness of a Hilbert space under an equivalent norm

Let us take the Hilbert space $l_2$ with an equivalent norm
$\Vert x \Vert = \max \{2 \Vert x \Vert_1, \Vert x \Vert_2 \}$, where $\Vert x \Vert_1 =( \sum_{n=2}^\infty x_n^2 )^{\frac{1}{2}}$ and $\...