Skip to main content

All Questions

Filter by
Sorted by
Tagged with
1 vote
2 answers
115 views

Computation of tangent functional

In Measures Which Agree on Balls by Hoffmann-Jørgenson, the tangent functional is defined as follows. If $x \in S$, we define the tangent functional $\tau(x,\cdot)$ at $x$ as \begin{equation} \...
1 vote
1 answer
343 views

Gateaux differentiability of the norm in Banach spaces

I'm struggling to understand a particular implication in the proof of Corollary 5 of this paper involving Gateaux differentiability of the norm. The claim is that Gateaux differentiability of the norm ...
0 votes
0 answers
145 views

$L_\infty([0,1], \mathbb{C})$ is it isomorphic to $\ell_\infty(\mathbb{N}, \mathbb{C})$?

By a result of Pełczyński, $L_\infty([0,1], \mathbb{R})$ is isomorphic to $\ell_\infty(\mathbb{N}, \mathbb{R})$. That is the case of real valued functions and sequences. A natural question then is: ...
3 votes
1 answer
161 views

Approximating continuous functions from $K\times L$ into $[0,1]$

Let $K$ and $L$ be compact Hausdorff spaces, let $f:K\times L\to [0,1]$ be continuous and let $\varepsilon>0$. Can we find continuous $g_{1},...,g_{n}:K\to[0,+\infty)$ and $h_{1},...,h_{n}:L\to[0,+\...
2 votes
0 answers
98 views

Geometric interpretation of uniform convexity condition

I first want to recall the moduli of uniform smoothness (US), uniform convexity (UC), asymptotic uniform smoothness (AUS), and asymptotic uniform convexity (AUC). Throughout, let $X$ be an infinite ...
6 votes
2 answers
4k views

Question about Schauder bases in C([0,1]).

I would like to check a statement about Schauder bases in $C([0,1])$ to be sure that I don't lie to my students on Monday. The statement(s) that I would like to check are: The family of monomials $\{...
3 votes
3 answers
776 views

Radon-Nikodym property for space of signed measures

Given a measurable space, the vector space of signed measures is a Banach space. Does it have the Radon-Nikodym property? What if the space is of a special type, such as a nice topological space with ...
5 votes
1 answer
221 views

Arens regularity of $\mathrm{BV}(\mathbb{R})$

$\DeclareMathOperator\BV{BV}$A Banach algebra $A$ is called Arens regular if the two canonical multiplications on the double dual $A^{**}$ coincide. Let $\BV(\mathbb{R})$ denote the Banach algebra of ...
4 votes
2 answers
1k views

Characterizations of Wiener algebra

The Wiener algebra $\mathcal W$ is defined as $\text{Fourier}(L^1(\mathbb R))$, i.e. the image by the Fourier transform of $L^1(\mathbb R)$. Riemann-Lebesgue's lemma ensures that $$ \mathcal W\subset ...
2 votes
1 answer
225 views

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 ...
19 votes
3 answers
1k views

What standard Banach space is isomorphic to the completion of this different normed structure on $\ell^1$?

A colleague asked me the following question: "What can one do with the following norm on $\ell^1$: $|x|=\int_1^2 |x|_pdp$ where $| \;\; |_p$ is the standard norm on $\ell_p$?" This ...
3 votes
1 answer
298 views

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 ...
1 vote
1 answer
310 views

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}}+...
0 votes
0 answers
80 views

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 ...
5 votes
1 answer
855 views

$L\log L$ 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 ...
2 votes
0 answers
78 views

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-...
0 votes
0 answers
137 views

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 ...
0 votes
0 answers
56 views

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’...
2 votes
1 answer
151 views

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\...
4 votes
0 answers
146 views

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) ...
0 votes
0 answers
168 views

Completely continuous maps from projective tensor products into $c_0$

Let $E$, $F$ be two Banach spaces and $E\mathbin{\hat{\otimes}}_{\pi}F$ denote their projective tensor product. For each unit norm $\xi\in E$ and $\gamma\in F$, let's define $$ J_{\gamma}:E\to E\...
0 votes
0 answers
47 views

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, \...
1 vote
0 answers
73 views

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, \...
1 vote
0 answers
50 views

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, |\...
0 votes
0 answers
113 views

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 ...
12 votes
0 answers
435 views

Uniform closure of subspaces of Baire class 1

Describe a uniformly closed linear subspace $A \subset C([0,1])$ such that the space $B_1(A)$ is not uniformly complete. Here $B_1(A)$ is the set of all bounded functions $f$ which are pointwise ...
1 vote
1 answer
264 views

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 ...
1 vote
0 answers
82 views

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 ...
0 votes
1 answer
165 views

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 ...
7 votes
1 answer
246 views

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 ...
7 votes
2 answers
419 views

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 $p > 1$. ...
6 votes
1 answer
167 views

Extension Operator for $W^{1,\infty}(U,X)$

I am reading through some lectures on Sobolev spaces and the vector-valued (or Banach space valued) version of them. At this moment I am very interested in extension operators for the vector-valued ...
0 votes
0 answers
77 views

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 ...
1 vote
0 answers
47 views

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_{\{||\...
0 votes
1 answer
145 views

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 $x=(x^1,x^2,\ldots)\in c_0$....
4 votes
1 answer
259 views

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} ...
0 votes
1 answer
141 views

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 ...
4 votes
2 answers
904 views

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 ...
2 votes
0 answers
79 views

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 ...
7 votes
1 answer
334 views

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 ...
2 votes
2 answers
167 views

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 ...
2 votes
1 answer
103 views

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)...
0 votes
0 answers
208 views

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 ...
2 votes
1 answer
321 views

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 ...
3 votes
0 answers
152 views

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 ...
1 vote
2 answers
310 views

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 $E_n$ ...
0 votes
1 answer
92 views

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. ...
1 vote
1 answer
113 views

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 ...
7 votes
1 answer
283 views

A characterization of Hilbert spaces by norm one projections

Suppose a (separable) Banach space $X$ has the following property: If $P:X\to X$ is a bounded projection different from $I$ such that $\|P\|=1$, then $\|I-P\|=1$. Does this imply that $X$ is a Hilbert ...
0 votes
0 answers
114 views

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 ...

1 2
3
4 5
25