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.

Filter by
Sorted by
Tagged with
-1
votes
0answers
57 views

$ h $ is an affine function?

Let $X$ be a separable Banach space, the associated dual space is denoted by $X^*$ and the usual duality between $X$ and $X^*$ by $\langle , \rangle$. For $C$ nonempty weakly compact convex subsets of ...
5
votes
1answer
82 views

quick question about renorming quasi-Banach spaces into p-Banach spaces

I have a quick question which is probably supposed to be obvious, but for some reason I just don't see it: How does one re-norm a quasi-Banach space to produce a $p$-Banach space ($0<p\leq 1$) ...
2
votes
0answers
19 views

A subclass of upper semi-Fredholm operators defined by the essential norms

For an operator $T:X\rightarrow Y$, we let $\|T\|_{e}$ denote the essential norm of $T$, that is, the distance from the compact operators, $$\|T\|_{e}:=\textrm{d}(T,\mathcal{K}(X,Y)).$$ We set $$\zeta(...
-1
votes
0answers
32 views

Aupetit B, A primer on spectral theory, Exercise IV.1 [closed]

This is my question in https://math.stackexchange.com/questions/3701351/aupetit-b-a-primer-on-spectral-theory-exercise- iv-1 I'm sorry reply here Aupetit B, A primer on spectral theory, Exercise IV....
1
vote
0answers
43 views

Can we say that $f$ admits a $m(X,X^*)$-continuous extension to $X$?

Let $X$ be a Banach space equipped with the Mackey topology $m(X,X^*)$. We suppose that $\big(X,m(X,X^*)\big)$ is separable space. Let $H$ be a countable, $m(X,X^*)$-dense subset with $(H=-H)$. Let $...
6
votes
1answer
67 views

If $X$ is separable space then $X^∗$ is separable in all topologies $\tau$ such that $(X^∗,\tau)^∗ =X$?

Let $(X,\|.\|_{X})$ be a separable Banach space and the associated dual space is denoted by $X^*$. By $w^*$ we shall indicate the weak$-*$ topology on $X^*$. Let $B_{X^∗}= \{x^∗ \in X^∗ : \|x^∗\|_{X^∗...
10
votes
0answers
177 views

Quantifier swap in Banach space theory

The uniform boundedness principle and its corollaries from a logical point of view are statements of when one can swap quantifiers in Banach spaces. Take for instance the principle of condensation of ...
0
votes
0answers
34 views

Schaten p norm of block matrices

Let $A=D\oplus 0$ be a diagonal Hermitian matrix and $B$ is an invertible Hermitian matrix with $(1,1)$ block being $B_{11}$ and $B_{11}$ and $D$ have the same dimensions. Then is it true that if $(1+|...
6
votes
3answers
764 views

Are nuclear operators closed under extensions?

Given $X_i, Y_i$ Banach spaces, $f_j, g_j, T_i$ bounded linear operators for $i=1,2,3$ and $j=1,2$. We have the following diagram $\require{AMScd}$ \begin{CD} 0 @>>> X_1 @>f_1>> X_2 ...
1
vote
0answers
43 views

Operational quantities characterizing upper semi-Fredholm operators

An operator $T:X\rightarrow Y$ is said to be upper semi-Fredholm if its range is closed and its kernel is finite-dimensional. M. Schechter (1972) introduced a quantity $$\nu(T):=\sup_{\operatorname{...
3
votes
1answer
58 views

Elementary relationships between finite-dimensional subspaces and finite-codimensional subspaces

I have an elementary question about finite-dimensional subspaces and finite-codimensional subspaces. This question may be known. Question. Let $U$ be a finite-dimensional subspace of an infinite-...
3
votes
0answers
95 views

Which metric spaces embed isometrically in $\ell_p$?

It is known that each metric space $X$ embeds isometrically in the Banach space $\ell_\infty(X)$ of bounded (not necessarily continuous) functions $X \to \mathbb R$. Since $\ell_\infty(X)$ does not ...
2
votes
0answers
83 views

Example of a non-reflexive Banach space and two sequences

Let $(E,\mathcal {A}, \mu ) $ be a finite measure space and $X$ be a Banach space. The set of all Bochner-integrable functions from $E$ into $X$ is denoted by $\mathcal{L}_X^1$. If $X$ is reflexive, ...
0
votes
1answer
98 views

$ \overline{(A-A)}\cap\overline{B}(0,r)\text{ is weakly compact, }\forall r>0 $?

Let $X$ be a separable Banach space and $A$ is a subset of $X$ such that $$ A\cap\overline{B}(0,r) \text{ is weakly compact, } \forall r>0. $$ Can we say that : $$ \overline{(A-A)}\cap\overline{...
6
votes
0answers
114 views

Quotients of subspaces of $C(\alpha)$

A well known problem, attributed to H. P. Rosenthal, asks whether or not every quotient of $C(\alpha)$, $\alpha$ countable ordinal, is $c_0$-saturated. As it is known, $C(\alpha)$ are $c_0$-saturated ...
4
votes
1answer
94 views

Existence of an injective unbounded below operator

Given an infinite dimensional Banach space, can we always find an infinite dimensional Banach space $Y$ and an injective bounded operator $T:X\to Y$ such that $T$ is not bounded below? If $X^{*}$ is ...
7
votes
1answer
340 views

Basis vs Schauder basis in normed spaces

Following the conventions from Heil: "A Basis Theory Primer" and Albiac, Kalton: "Topics in Banach Space Theory", we might define a basis of an (infinite-dimensional) normed space $V$ as a sequence $(...
5
votes
1answer
232 views

Existence of injective compact operators

We know that if $X$ is a separable Banach space, then for every infinite dimensional Banach space $Y$, there exists an injective compact operator from $X$ to $Y$. My query is for every Banach ...
2
votes
2answers
140 views

Can we say that : $ (A-B)\cap\overline{B}(0,r)\text{ is weakly compact, }\forall r>0 $

Let $X$ be a separable Banach space and $A,B$ are closed convex subsets of $X$ such that $B\subset A$ and $$ A\cap\overline{B}(0,r) \text{ and } B\cap\overline{B}(0,r) \text{ are weakly compact, } \...
7
votes
0answers
91 views

The bidual of the space of divergence-free vector fields

Consider the Banach space $L_1(\mathbb R^n, \mathbb R^n)$ of integrable vector fields $(n>1$) together with its subspace $N$ formed by those vectors fields whose divergence (computed in the ...
2
votes
1answer
101 views

complemented subspace of the direct sum of two Banach spaces

When I was reading a paper, I saw something like: If $F$ and $E$ are Banach spaces with symmetric bases (precisely, they are symmetric sequence spaces), and $F$ is isomorphic to a complemented ...
0
votes
0answers
44 views

Closed graph theorem for cones?

In the paper "A strong open mapping theorem for surjections from cones onto Banach spaces, Marcel de Jeu and Miek Masserschmidt, Adv. Math." it is proved (among other things) that if $X, Y$ are ...
4
votes
2answers
187 views

Injective continuous operators between Banach spaces

Suppose $X$ and $Y$ are two infinite dimensional Banach spaces. What can we say about the set of all injective continuous linear operators between $X$ and $Y$? Is it always nonempty?
6
votes
1answer
94 views

closed ideals in L(L_1)

Denote $L_1=L_1[0,1]$ The lattice of closed ideals in $\mathcal{L}(L_1)$ includes the chain $$ \{0\}\subsetneq\mathcal{K}(L_1)\subsetneq\mathcal{FS}(L_1) \subsetneq\mathcal{J}_{\ell_1}(L_1)\subsetneq\...
-1
votes
2answers
125 views

inequivalent norms [closed]

I am thinking about the following question: Let $X$ be a Banach space, say separable, e.g., $l_p$ or $c_0$. When can I say that there exist inequivalent complete norms on $X$?
2
votes
0answers
61 views

Comparison of inductive limit topology with very-rapidly decaying non-convex $L^{!/2}$-space topology

This is related to these posts and here. Let $L^1([n,n+1])$ denote the subspace of $L^p$-functions on $[0,\infty)$ essentially supported on $[-n,n]$. Denote the accelerated $\ell^1$-direct sum ...
1
vote
0answers
137 views

Use of this space of very rapidly decreasing continuous functions

Let $C_n$ denote the subspace of continuous function on $[0,\infty)$ supported on $[n,n+1]$. Denote the $\ell^p$-direct sum Banach space $$ V_p := \left\{ f \in C([0,\infty)):\, \sum_{n=1}^{\infty} ...
2
votes
1answer
183 views

Separable Banach spaces isometric to quotient of a Banach space

We know that every separable Banach space is isometrically isomorphic to a quotient space of $(\ell^1,\|.\|_1)$. We also know that the norm defined by $\|x\|=(\|x\|_1^2+\|x\|_2^2)^{1/2}$ for all $x\in ...
1
vote
1answer
70 views

Convergence in LB-spaces

Edit: Let $X$ be a strict LB-space described by $\lim X_n$ and suppose that $\{x_n\}_{n \in \mathbb{N}}$ converges in $X$. I'm looking for a reference showing that $x_n$ must converge in some $X_N$.
3
votes
1answer
150 views

Weak convergence in $L^1(X,\mu)$ space

I found an interesting property in some lecture notes on weak convergence: lemma 3.2 (2) on page 10: https://www.uio.no/studier/emner/matnat/math/MAT4380/v06/Weakconvergence.pdf . The idea is as ...
10
votes
1answer
244 views

Equivalence of σ-convex hull and closed convex hull

Let $X$ be a locally convex topological space, and let $K \subset X$ be a compact set. Recalling that the standard convex hull is defined as $$\text{co}(K) = \Big\{ \sum_{i=1}^n a_i x_i : a_i \geq 0,\,...
4
votes
1answer
57 views

Two measures of noncompactness of operators

Let $X,Y$ be Banach spaces. We denote by $\mathcal{K}(X,Y)$ the space of all compact operators from $X$ into $Y$. For an operator $T:X\rightarrow Y$, we let $$\|T\|_{e}:=\inf\{\|T-K\|:K\in \mathcal{K}(...
6
votes
1answer
272 views

Cartesian product of Banach spaces: all norms such that the inclusion is an isometry are equivalent?

Let $\mathcal{A}$ be an arbitrary (typically infinite-dimensional) Banach space with norm $\|\cdot\|_{\mathcal{A}}$ and let $\mathcal{A}^{n}$ be its Cartesian product. I came across the following ...
2
votes
0answers
69 views

A quantitative characterization of bounded approximation property

Recall that a Banach space $X$ has the approximation property (AP for short ) if for every compact subset $K$ of $X$ and every $\varepsilon > 0$, there exists a finite rank operator $S$ on $X$ such ...
4
votes
0answers
73 views

Closed subgroup (Cartan) theorem without transversality nor Lipschitz condition within Banach algebras

Yesterday, I came across the following preliminary theorem. Theorem Let $\mathcal{B}$ be a Banach algebra (with unit $e$) and $G$ be a closed subgroup of $\mathcal{B}^{-1}$ (the group of ...
8
votes
2answers
202 views

On dense embedding of Banach spaces

Disclaimer: When I came up with this question yesterday, I suspected it to be trivial (trivially true or trivially false). Then it kept me awake several hours tonight... (I still hope, though, this is ...
10
votes
1answer
285 views

Is the spectrum of a “self adjoint” operator real on $\ell^p$?

There might be an obvious answer to the question, but it doesn't come to mind. Suppose we have an infinite matrix $A=(a_{ij})$, which defines a bounded linear operator on $\ell^p$, i.e. for all ...
7
votes
2answers
194 views

Completeness of coefficient functionnals

My questions is about Schauder bases and more specifically about coefficient functionals. Let $(x_n)$ be a Schauder basis of a Banach space $X$. Thus for all $x$ in $X$, $x = \sum f_n(x) x_n$. The $...
7
votes
0answers
192 views

Isometries on the unit sphere

Suppose that $X$ and $Y$ are two Banach spaces, $S_{X}$ and $S_{Y}$ their unit spheres, and $f$ an onto isometry between $S_X$ and $S_Y$. Does it follow that $X$ and $Y$ are isometric?
6
votes
1answer
63 views

Containment of $c_0$ in projective tensor products

Let $X$ and $Y$ be Banach spaces and denote by $X\hat{\otimes}_\pi Y$ the projective tensor product. Question: If $X\hat{\otimes}_\pi Y$ contains an isomorphic copy of $c_0$, must then $X$ or $Y$ ...
3
votes
1answer
124 views

Reference request: completion of Banach norm on sum

Let $X_1,X_2$ be Banach subspaces of a locally convex space $X$. Then the subset $$ X_1+X_2 = \left\{ x\in X:\, x= \beta_1 x_1 + \beta_2 x_2 \, \beta_i \in \mathbb{R},\, x_i \in X_i \right\}, $$ a is ...
6
votes
1answer
106 views

Density and the projective tensor product

Let $X$ be a locally convex space (over $\mathbb{R}$), $D\subset X$ be dense, $B$ be a Banach space (again over $\mathbb{R}$) with Schauder basis $\{b_i\}_{i =1}^{\infty}$. Is the set $$ D^+\...
0
votes
1answer
80 views

Law of a step function and its generalization to two dimensions on an appropriate spaces

Let's consider two discontinuous functions defined on $D$ and $D \times [0,T]$, respectively: A step function: $u_1(x)=\begin{cases} u_{L}, x<c_1, \\[2ex] u_{R}, x>c_1, \end{cases}$ A "...
8
votes
0answers
176 views

Von Neumann Inequality in Banach spaces

It is known that the only Banach space that satisfies the von-Neumann inequality is the Hilbert space: Theorem (see e.g. Pisier, "Similarity Problems and Completely Bounded Maps", p 27) For a Banach ...
0
votes
1answer
61 views

Example of a nonconvex Chebyshev set in a metric space with continuous projection?

Question: Is there an example of a nonconvex Chebyshev set $S$ in a metric space $(X,d)$ whose projection map is continuous? For convexity to be well-defined, we need to assume that $X$ is a vector ...
2
votes
0answers
65 views

Banach lattices $X$ for which $L_p(\mu)\subset X$ or $X\subset L_q(\mu)$

It is well known (see vol. II of Lindenstrauss and Tzafriri's book) that an order continuous Banach lattice $X$ with a weak unit admits a representation as a (in general not closed) ideal of $L_1(\mu)$...
0
votes
1answer
36 views

A closed subspace generated by open $F_{\sigma}$ sets of $K$ in $C(K)^{**}$

Let $K$ be a compact Hausdorff space. For a bounded Borel measurable function $f$ on $K$, we define $\phi_{f}\in C(K)^{**}$ by $\phi_{f}(\mu)=\int_{K}fd\mu$ for all $\mu\in C(K)^{*}$. It is easy to ...
3
votes
1answer
261 views

Higher order functional derivatives

Let $E, F$ be Banach spaces. A continuous bilinear functional ${\langle \cdot\,, \cdot \rangle }: E \times F \to \mathbb{R}$ is called $E$-non-degenerate if $\langle x,y\rangle = 0$ for all $y \in F$ ...
4
votes
1answer
76 views

A subspace generated by closed $G_{\delta}$ sets in $K$ between $C(K)$ and $C(K)^{**}$

Let $K$ be a compact Hausdorff space. We let $Z$ be the closed subspace generated by $\{\chi_{F}:F$ closed $G_{\delta}$ sets in $K\}$ in $C(K)^{**}$. My question is the following: Question 1. Is $C(K)...
1
vote
1answer
55 views

Metrization of quotient spaces defined by sequences of continuous functions

Let $K$ be a compact Hausdorff space and let $C(K)$ be the space of all scalar-valued continuous functions on $K$. Let $(f_{n})_{n}$ be a sequence in $C(K)$ satisfying $\sup\limits_{n}\sup\limits_{t\...

1
2 3 4 5
23