# 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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questions

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### $ 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 ...

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### 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 ($0<p\leq 1$) ...

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### 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(...

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### 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....

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### 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 $...

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**1**answer

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### 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^∗...

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### 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 ...

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### 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+|...

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### 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 ...

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### 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{...

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**1**answer

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### 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-...

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### 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 ...

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### 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, ...

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**1**answer

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### $ \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{...

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### 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 ...

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**1**answer

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### 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 ...

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**1**answer

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 $(...

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**1**answer

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 ...

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### 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, } \...

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### 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 ...

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**1**answer

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### 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 ...

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### 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 ...

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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?

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### 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\...

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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 $X$?

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### 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 ...

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### 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} ...

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**1**answer

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 ...

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### 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 $X_N$.

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### 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 ...

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### 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,\,...

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### 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}(...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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**1**answer

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### 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 ...

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### 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 $...

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### 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?

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### 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 $Y$ ...

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**1**answer

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### 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 ...

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**1**answer

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### 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^+\...

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### 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 "...

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### 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 ...

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**1**answer

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### 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 ...

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### 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 $L_1(\mu)$...

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**1**answer

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### 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 ...

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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 $y \in F$ ...

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**1**answer

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### 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)...

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### 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\...