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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Combination of simple tensors - II

This is a follow-up question to Combination of simple tensors. I am interested in devising an alternative norm (I mean, other than the usual $\pi$ or $\epsilon$ norms) in the tensor product of two ...
Lorenzo Guglielmi's user avatar
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Minimal norm problem with linear combination of translation operator to be estimated

Follow up question from this one Suppose $X = L^2(G)$, where $G$ is some locally compact group. Let $x, y \in G$ I for fixed $n$ I am seeking for an operator $H \in B(X)$ of the form $$ H = H(\alpha_1,...
user8469759's user avatar
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On a property for normed spaces

I asked this question on Math Stackexchange, but I didn't get an answer: https://math.stackexchange.com/questions/4881155/on-a-property-for-normed-spaces?noredirect=1#comment10410489_4881155 I came ...
Markus's user avatar
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Strong sub-differentiability of an equivalent strictly convex norm

First, we define the notion of strong sub-differentiability(SSD) of a norm on a Banach space $X$. The norm $\Vert \cdot \Vert$ of $X$ is said to be SSD if the one-sided limit $$\lim_{t \to 0+} \frac{\...
PPB's user avatar
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Combination of simple tensors

I aksed this question on Math Stack Exchange 6 days ago, with no answer: https://math.stackexchange.com/q/4875445/1297919 Let $X$ and $Y$ two Banach spaces and let $X\otimes Y$ their tensor product. ...
Lorenzo Guglielmi's user avatar
2 votes
1 answer
201 views

Banach spaces locally having a basis

The $\mathcal{L}_p$-spaces ($1\leq p \leq \infty$) are Banach spaces $X$ such that there exists a constant $\lambda$ so that every finite dimensional subspace $E$ of $X$ is contained in another ...
M.González's user avatar
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Convergence of slice in an equivalent renorming

Let us consider $\ell_2$ space with $\Vert \cdot \Vert_2$ norm. Let us define a new norm equivalent to $\Vert \cdot \Vert_2$ norm as follows: $$ \Vert x \Vert_0 = \max \{ \Vert x \Vert_2, \sqrt{2} \...
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Are projective tensor products left-exact if one considers only maps of norm at most 1?

Consider the category $\mathrm{Ban}$ of Banach spaces and bounded linear maps and the category $\mathrm{Ban}_1$ of Banach spaces and bounded linear maps of operator norm at most 1. Let $\otimes_\pi$ ...
Stephan Mescher's user avatar
2 votes
1 answer
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Are these two norms on localized versions of $L^p_q$ equivalent?

$\newcommand{\RR}{\mathbb R}\newcommand{\diff}{\, \mathrm d}$ We fix $T \in (0, \infty)$ and $p, q \in [1, \infty)$. Let $\mathbb T$ be the interval $[0, T]$. Let $E$ be the space of all real-valued ...
Akira's user avatar
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Convexity property of an equivalent norm on $\ell_2$

Let us consider the space $\ell_2$ with an equivalent norm defined by $$ \Vert x \Vert = \max \{ \Vert x^{'} \Vert_2, \Vert x^{''} \Vert_2 \}, $$ where $x^{'}=(0, x_2, x_3, \cdots)$, $x^{''} = (x_1, 0,...
PPB's user avatar
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6 votes
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If a Banach / Fréchet manifold $M$ happens to be a topological vector space, is $M$ just a Banach / Fréchet space?

In finite dimensions, if $M$ is a smooth manifold that happens to be a vector space, then it is indeed just the Euclidean space. I wonder if the same result holds valid in infinite dimensions. More ...
Isaac's user avatar
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Find reasonable definition for endpoint Lorentz function spaces $L^{\infty,q}$ via the idea from endpoint Triebel-Lizorkin ${\scr F}_{\infty,q}^s$

On a measure space $(X,\mu)$, for $0<p,q<\infty$ the Lorentz space $L^{p,q}(\mu)$ is defined by $$\|f\|_{L^{p,q}(\mu)}:=p^\frac1q\|t\mu(|f|>t)^\frac1p\|_{L^q(\mathbb R_+,\frac{dt}t)}=p^\...
Liding Yao's user avatar
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Does there exists an example of a Banach space that is compactly LUR; but not LUR

We know that a Banach space $X$ is locally uniformly rotund(LUR) if for $x, x_n \in S_X$ with $\Vert x+x_n \Vert \to 2$, we have $x_n \to x$. In the same case, if $(x_n)$ has a convergent subsequence, ...
PPB's user avatar
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Weakly null sequences in projective tensor products II

The question in this post is the question below from an article by Rodriguez & Rueda Zoca [1]. Below is a complimentary salad/side dish that accompanies the main course. Let $B^2(X,Y)$ denote ...
Onur Oktay's user avatar
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4 votes
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Large JN-sets in Banach spaces

For every infinite-dimensional Banach space $X$ there is a weak*-null sequence in the unit sphere of $X^\ast$. Does this extend under suitable circumstances to the non-separable setting? Say that $X$ ...
user340234's user avatar
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Is such a Banach space $X$ isometrically isomorphic to a Hilbert space or not?

Let $X$ be a real or complex Banach space. $X$ satisfies:There exists real or complex series $\{a_k\}_{k=1}^n,\{b_k\}_{k=1}^n$ (which satisfies that:$\begin{cases} a_k,b_k\in \mathbb{R}\ \forall k=1,\...
anyon's user avatar
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Array-determined operator ideals

For a Banach space $X$, we, of course, know what it means for a sequence to be weakly null (to converge to zero in the weak topology). An array in the Banach space $X$ is a sequence of sequences, $(...
jwhite's user avatar
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Reconstructing the manifold from space of functions in quantum mechanics

Due to Banach–Mazur, every separable Banach space is isomorphic to a subspace of $C([0,1])$. But some spaces, like $C([0,1]^n)$ and generally $C(M)$ for $M$ a manifold, allow one to reason about the ...
0x11111's user avatar
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6 votes
2 answers
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Taylor expansion theorem for Gateaux differentiable functions

I am having a hard time studying Gateaux derivatives (see https://en.wikipedia.org/wiki/Gateaux_derivative), it seems that every author mentions the concept but only as a cliffhanger to study Fréchet ...
Antonio Martins Alves Veloso d's user avatar
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Isomorphic copies of $c_0$ in the projective tensor products

There exist Banach spaces $X$ such that the projective tensor product $X\mathbin{\hat{\otimes}}_\pi X$ contains an isomorphic copy of $c_0$ [BourgainPisier1983]. Moreover, $X$ is an $\mathcal{L}_\...
Onur Oktay's user avatar
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7 votes
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Using the Stone-Weierstrass theorem to solve an integral limit

The following question was posted on math stack exchange here but it got no answers Let $c\in (1, +\infty)$ and $f \colon [0, c] \to \mathbb{R}$ be a continuous and monotonically increasing function ...
Shthephathord23's user avatar
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Density of the set of convex polygons in the Banach-Mazur distance

Is the set of convex polygons dense in the set of convex domains in $\mathbb{R}^2$, for the Banach-Mazur distance? Any insight for a negative or positive answer is very much welcome!
kvicente's user avatar
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Weak$^\ast$ closure of a countably complete sublattice of the unit ball of $L^\infty(\Omega, \mu)$

This is a reframing of my previous question from a Banach lattice perspective: Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity? The previous question ...
David Gao's user avatar
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1 answer
155 views

Differentiation of a norm

I am interested in determining whether the function $F:L^{\infty}(0,\infty;L^{\infty}(0,1)) \to \mathbb R$ defined by $$u \mapsto F(u)=\int_0^\infty \int_0^1 u(x,t)^2 \ \mathrm dx \, \mathrm dt $$ is ...
elmas's user avatar
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Why do distributional isomorphisms preserve joint distribution?

Let $(\Omega,\mathcal{A},\mu)$ and $(\Omega',\mathcal{A}',\mu')$ be probability spaces and $$f_1,\ldots,f_n:\Omega\to\mathbb R,\; f_1',\cdots, f_n':\Omega'\to\mathbb{R}$$ be integrable random ...
Pavlos Motakis's user avatar
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Non-degenerate representation of a Banach algebra

Let $\mathcal{A}$ be a non-reflexive Banach algebra. For the definition of Arens product, please refer to this link. Here we let $\square$ denote the first Arens product and $\diamond$ denote the ...
Sanae Kochiya's user avatar
1 vote
1 answer
242 views

An example of non-invertible operator $F$ such that $P_nF$ is invertible on $\operatorname{Im}P_n$ or proving that It is impossible

Given: $X$ - any Banach space $F : X \to X$ (linear bounded and non-invertible) $P_n$, which is projector that strongly converges to the identity operator $I$ as $n \to\infty$ Can you help me come ...
TorteDeline's user avatar
4 votes
0 answers
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On the predual of the James tree space $\mathit{JT}$

$\newcommand\JT{\mathit{JT}}$The James tree space $\JT$ was the first example of a separable Banach space containing no copies of $\ell_1$ such that its dual space is non-separable. Since $\JT$ admits ...
M.González's user avatar
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When do the weak-star and compact convergence (compact-open) topology coincide on the dual of a Banach space?

In Measures Which Agree on Balls by Hoffmann-Jørgenson, it is claimed on page 323 that for an arbitrary Banach space $E$, if $\pi$ is the topology on $E^*$ of uniform convergence on compact subsets of ...
i like math's user avatar
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When can an affine functional on the dual be represented as an element of a Banach space?

In Measures Which Agree on Balls by Hoffmann-Jørgenson, we are given a functional $\varphi: T(x_0)\to (-\infty, \infty]$, which is a lower semicontinuous, affine, Baire function on a subspace $T(x_0)$ ...
i like math's user avatar
1 vote
2 answers
110 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} \...
i like math's user avatar
1 vote
1 answer
178 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 ...
i like math's user avatar
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$L_\infty([0,1], \mathbb{C})$ 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: ...
NotaChoice's user avatar
1 vote
1 answer
82 views

Potentially elementary question on affine functions on Banach spaces

In Measures Which Agree On Balls by Hoffmann-Jørgensen, it is claimed that the function defined on $T(x)$, the set of normals to the unit sphere at $x$, given by $ \varphi(x^*) = \left\{ \begin{array}{...
i like math's user avatar
3 votes
1 answer
145 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,+\...
erz's user avatar
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2 votes
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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 ...
user516424's user avatar
1 vote
1 answer
158 views

Definition and properties of tangent functional

I am reading Measures Which Agree on Balls by Hoffmann-Jørgensen and I am somewhat confused. Here, $E$ is a Banach space, $S$ is the unit sphere, and $x \in S$. We let $\tau(x, \cdot)$ denote the ...
i like math's user avatar
13 votes
2 answers
698 views

Smooth Urysohn's lemma on Fréchet spaces

Let $V$ be a Fréchet topological vector space. Let $K_0$ and $K_1$ be two closed subsets which are disjoint. I wish to show the existence of a Fréchet-smooth function $f:V\to [0,1]$ whose restriction ...
André Henriques's user avatar
11 votes
0 answers
242 views

Continuous representations of topological groups on Banach spaces

Let $G$ be a topological group and let $\rho:G\rightarrow\text{GL}(V)$ be a linear representation of $G$ on a Banach space $V$. The representation is called strongly continuous if the map $g\mapsto\...
Botwinnik's user avatar
2 votes
1 answer
274 views

interiors of positive cones in ordered Banach spaces

I have a couple of questions about ordered Banach spaces and interiors of their positive cones. I would appreciate your insights and any recommended references. I want to know several examples of ...
Saito's user avatar
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1 vote
1 answer
125 views

Smoothness of an equivalent norm

For an arbitrary set $\Gamma$, Day's norm on $c_0(\Gamma)$ is defined by $$ \Vert x \Vert = \sup \bigg \{ \bigg ( \sum_{k=1}^n 4^{-k} x^2(\gamma_k) \bigg )^{\frac{1}{2}} : (\gamma_1, \cdots, \gamma_n) ...
PPB's user avatar
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5 votes
1 answer
181 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 ...
Tobias Fritz's user avatar
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1 vote
0 answers
174 views

Trans-universality for finite-dimensional Banach space

In addition to a specific problem Trans-universality for finitely generated groups, I posted also its general form. It should not hurt to provide another special case: QUESTION: does there exist a ...
Wlod AA's user avatar
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2 votes
1 answer
221 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 ...
Sanae Kochiya's user avatar
1 vote
1 answer
134 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}}+...
InMathweTrust's user avatar
0 votes
0 answers
60 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 ...
Martin Geller's user avatar
4 votes
1 answer
170 views

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\...
Bogdan's user avatar
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3 votes
1 answer
285 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 ...
erz's user avatar
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1 vote
0 answers
64 views

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 ...
0x11111's user avatar
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2 votes
0 answers
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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-...
Onur Oktay's user avatar
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