**0**

votes

**0**answers

30 views

### A nullspace identity for operator exponentials

Let $X$ be a complex Banach space. Does validity of
$$
\mbox{ker}\left(e^{2\pi \imath \, T} - 1\right) = \sum\nolimits_{k\in \mathbb{Z}} \mbox{ker} (T-k) \quad \forall \, T \in B(X,X)
$$
imply that ...

**0**

votes

**0**answers

33 views

### interpolation between Bochner spaces

Is there a reference for the interpolation result stating the existence of an embedding
\begin{equation}
L^2(I;W^{2,p}(\Omega)) \cap H^1(I;L^p(\Omega)) \hookrightarrow ...

**3**

votes

**0**answers

67 views

### independent symmetric 3-valued random variables in Lp

Consider the following excerpt from this paper:
Given $1<p<2$, $0<w\leq 1$ and $n\in\mathbb{N}$, we fix once and for all a sequence $f_j^{(n)}=f_{p,w,j}^{(n)}$, $1\leq j\leq n$, of ...

**1**

vote

**0**answers

75 views

### What is the trace of this operator in $L^\infty$ (if this question make sense)?

Let $\Omega$ be a closed domain in $\mathbb{R}^N$, and $\lambda$ the corresponding Lebesgue measure. We define in $(L^\infty(\Omega), \Vert \cdot \Vert_\infty)$ the following operator :
...

**2**

votes

**1**answer

98 views

### Rearrangments of Fourier series

Suppose one has a schauder basis $\{f_n\}_{n\in\mathbb{N}}$ for $L^p([0,1])$ and we wish to expand a function $f \in L^p([0,1])$ in our basis to get the expression
$$f(y)=\sum_{n=0}^{\infty} a_n ...

**0**

votes

**0**answers

35 views

### Bounded linear functionals over smooth maps of a compact interval [migrated]

I have two questions regarding the topological dual of the space $E = \mathcal{C}^\infty([0; 1])$ of infinitely continuously differentiable functions over the closed interval $[0; 1]$ equipped with ...

**1**

vote

**1**answer

81 views

### A question on p-summing operators

Let $(x_{n})_{n}$ be a $p$-summable sequence in a Banach space $X$. Define an operator $T$ from $l_{q}$(where $q=p/(p-1)$) to $X$ by $Te_{n}=x_{n}(n=1,2...)$. Is $T$ a $p$-summing operator?

**-1**

votes

**0**answers

44 views

### On $p$-summable sequences with respect to operator ideals

We say that a sequence $(x_{n})_{n}$ in a Banach space $X$ is $p$-summable with respect to a Banach operator ideal $(\mathcal{A},\alpha)$ if there exist a Banach space $Z$, an operator ...

**2**

votes

**0**answers

67 views

### When do Borel $\sigma$-algebras generated by the total variation norm and the weak* topology coincide?

I am almost certain that I read somewhere that the following is true, but I cannot seem to locate the reference. I would be most appreciative if someone could point me to a reference. The result was ...

**3**

votes

**1**answer

79 views

### Banach-Stone Theorem in Lipschitz-free spaces

If $T$ is a nonlinear surjective isometry from Lipschitz-free space $\mathcal{F}(M)$ to $\mathcal{F}(N)$($M,N$ are metric spaces), is $M$ homeomorphic to $N$?

**4**

votes

**1**answer

181 views

### Open problems in Banach spaces, universality

I have gathered a list of universality problems in Banach spaces which have been solved:
1.The non existence of a separable reflexive space universal for the class of separable reflexive spaces.
...

**4**

votes

**0**answers

111 views

### SubGROUPs of Banach spaces, when are they dense in a vector subspace?

It’s relatively easy to show that if $J$ is a closed subgroup of a finite-dimensional real Banach space, $B$, then it is a vector subspace iff for all bounded linear functionals $\sigma$ of $B$, ...

**1**

vote

**0**answers

104 views

### Matrix norm on $L^p$ operator algebras

I have a question about the framework of $L^p$ operator algebras ($1<p<\infty$), i.e. norm-closed subalgebras of $B(L^p(X,\mu))$ for some measure space $(X,\mu)$ (see e.g. ...

**3**

votes

**1**answer

113 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 ...

**5**

votes

**1**answer

174 views

### Location of a Banach Space inside its bidual

Let $X$ be a Banach Space and let $Y$ be a closed subspace of $X^{**}$ such that $X\bigcap Y=0$. Let $P$ be the quotient map from $X^{**}$ onto $X^{**}/ Y$. I need to prove or refute that ...

**12**

votes

**1**answer

512 views

### Are “most” operators on an infinite-dimensional complex Banach space “diagonalizable”?

This is true for finite-dimensional spaces: the diagonal operators on a finite dimensional complex vector space form contain a dense open set and the nondiagonalizable operators have measure 0.
To be ...

**1**

vote

**0**answers

99 views

### Estimates of entropy of functional spaces

Let $M^n$ be a compact $n$-dimensional manifold. For $k\geq 0$ let us denote by $C^k(M)$ the Banach space of $k$ times continuously differentiable functions, and $B_{C^k}$ denote the unit ball of it.
...

**3**

votes

**2**answers

148 views

### Basis equivalent with a monotone basis

Given a basis in a Banach space $X$, can one find, for every $\varepsilon>0$, an equivalent basis with basis constant at most $1+\varepsilon$?
In $L_p[0,1]$ with $1<p<\infty$ any monotone ...

**9**

votes

**4**answers

415 views

### Between Tietze's and Dugundji's Extension Theorems

The celebrated Tietze Extension Theorem asserts that any continuous real-valued function defined on a closed subset of a normal space, can be extended to a continuous function on the whole space. Seen ...

**4**

votes

**3**answers

544 views

### reflexive banach space

I want to ask this non-expert question:
What does it mean geometrically for a Banach space to be reflexive?
Well, we could say a Banach space is reflexive iff unit ball is weakly compact. Or some ...

**3**

votes

**1**answer

187 views

### A question in Banach space

Given $A\in\Bbb R^{m\times n}$, define $$\gamma_2(A)=\min_{XY=A}\max_{i,j}\|x_i\|_{\ell_2}\|y_j\|_{\ell_2}$$ where rows of $X$,columns of $Y$ given by $\{x_i\}_{i=1}^m,\{y_j\}_{j=1}^n$ respectively.
...

**7**

votes

**0**answers

185 views

### Lipschitz-free spaces of $\mathbb R^n$

We define
$$
\text{Lip}_0(\mathbb R^n)=\{f:\mathbb R^n\rightarrow \mathbb R, \text{such that $f(0)=0$ and }
\sup_{x\not=y}\frac{\vert f(x)-f(y)\vert}{\vert x-y\vert}<+\infty.
\}
$$
It is well-known ...

**2**

votes

**2**answers

107 views

### Infinite direct sum of $l_2^{(n_k)}$ contains a complemented isometric copy of $l_2$

How do I show that for any increasing sequence $(n_k) \subseteq \mathbb{N}$, the space $\left( \oplus _{k=1} ^\infty l_2 ^{(n_k)} \right) _\infty$ contains a complemented isometric copy of $l_2$?

**0**

votes

**0**answers

96 views

### Banach space dual to $L^\infty(I,H^1(M))$

What is the dual to $L^\infty (I,H^1(M))$?, where $I$ is an interval in the real line; $H^1(M)$ is Sobolev space of degree 1, and $M$ is a compact manifold like the torus.
Any references that show ...

**2**

votes

**1**answer

116 views

### Predual of a subspace

Let $E$ be a Banach space, let $d\ge 1$ be an integer. Let $\mathcal G$ be a weakly closed subspace of
$(E^*)^d$ with finite codimension.
I would like know if the space $\mathcal G$ is a dual space ...

**1**

vote

**0**answers

122 views

### Inductive/Projective Limits of Topological Algebras

It is common to form inductive/projective limits of Banach/Frechet spaces in order to come up with natural topologies for common vector spaces. For instance,
For $k \ge 0$ and $K_n$ compact ...

**15**

votes

**2**answers

433 views

### Ultraweak topology on B(X): Is the map X\otimes X* -> B(X)* isometric?

Let $X$ be a Banach space. Consider the map
$$
\alpha\colon X\hat{\otimes} X^* \to B(X)^*,
$$
defined one simple tensors as
$$
\alpha(\xi\otimes\eta)(a) = \eta(a(\xi)).\quad (\xi\in X, \eta\in X^*, ...

**2**

votes

**1**answer

144 views

### Is this structure a Banach bundle?

Let $X$ be a Banach space. Put $Y=\{ \phi\in X^{*}\mid\;\; \parallel \phi \parallel\leq 1\;\; \&\;\; \phi \neq 0\}$ which is a locally compact Hausdorf space with the weak star topology.
...

**3**

votes

**0**answers

90 views

### A question on metric characterization of approximation property

My question is: a Banach space $X$ has the approximation property if and only if for every $\epsilon>0$ and every compact subset $K\subset X$, there exists a Lipschitz map $T: X\rightarrow X$ with ...

**3**

votes

**4**answers

629 views

### Non-separable Banach space

The vector space $C_b(\mathbb R)$ of bounded continuous functions on $\mathbb R$ is non-separable: it is possible to produce a direct proof of this fact, mimicking the standard proof for the ...

**6**

votes

**1**answer

451 views

### Are $L^\infty(\Bbb R)$ and $L^2(\Bbb R)$ homeomorphic?

It's easy to see that, for $1\le p,q< \infty$ the spaces $L^p(\Bbb R)$ and $L^q(\Bbb R)$ of $p$-th and $q$-th power integrable functions on the real line are homeomorphic as topological spaces. In ...

**1**

vote

**1**answer

109 views

### Approximation Property: Decomposition

This thread originated from MSE: Approximation Property: Decomposition
Given a Banach space $E$.
Consider a finite rank operator $F\in\mathcal{F}(X,E)$.
Introduce a basis on the finite dimensional ...

**0**

votes

**0**answers

73 views

### Approximation Property: Characterization

Problem
Given a Banach space $E$. Denote compact sets by $\mathcal{C}$, compact operators by $\mathcal{C}(X,Y)$, and finite rank operators by $\mathcal{F}(X,Y)$.
Suppose it has the approximation ...

**0**

votes

**0**answers

72 views

### Compact Approximation

This thread originated from MSE: Compact Approximation
This is meant as lemma for: Approximation Property
Given a Banach space $E$.
Denote compact domains by $\mathcal{C}$.
Denote compact ...

**5**

votes

**1**answer

308 views

### Non-reflexive Banach space s.t. X,X*,X**,… are separable

Is there an infinite-dimensional Banach space $X$, which is not reflexive, such that all the spaces $X,X^{\ast},X^{\ast\ast}, X^{\ast\ast\ast},\dots$ are separable?

**0**

votes

**0**answers

86 views

### Can we conclude that $\varphi(L_xf_0)\neq0$ for every $x\in G$?

Let $H$ be a compact subgroup of locally compact topological group $G$ and $A=\{f\in L^1(G); R_hf=f(a,e)\forall h\in H\}$ as a subalgebra of $L^1(G)$ by convolution of $L^1(G)$. If $\varphi \in ...

**7**

votes

**1**answer

149 views

### Dual of Banach-valued $L^p$ [duplicate]

Let $X$ be an infinite-dimensional Banach space and let $p\in(1,+\infty)$. We may define $L^p(\mathbb R;X)$. Is it always true that the topological dual of $L^p(\mathbb R;X)$ is $L^{p'}(\mathbb ...

**0**

votes

**0**answers

48 views

### Completeness of spaces $\Lambda(\varphi, p)$

Definition 1. Let $f$ be measurable function on a mesurable subset $E\subset \mathbb R^n$. Non-increasing rearrangement of $f$ is a function $f^\ast(x)=\inf\{s>0: \operatorname{mes} E[|f| > s] ...

**7**

votes

**0**answers

106 views

### Approximate singular value decomposition in Banach spaces

I am interested in generalisations to Banach spaces of the following construction, which relates to the singular value decomposition of a finite-dimensional linear map. If $V$, $W$ are ...

**1**

vote

**0**answers

71 views

### A reasonable framework to study properties of operator $A \mapsto KAK$ on Banach space

Let $K$ be a continuous linear operator on $C[0,1]$ (more, precisely, it is a linear integral operator). Then $K$ defines a continous linear operator $\widehat K$ on $\mathcal L(C[0,1])$ by the rule
...

**2**

votes

**1**answer

210 views

### Integration in C^* algebra

Let $\mathfrak{A}$ be a C${}^*$ algebra and $\mathbb{R}\ni s \mapsto \alpha_s$ a continuous family of its automorphisms. Is it true that
$$
\int d s \, f(s)\, \alpha_s(A)
$$
is well defined as a ...

**3**

votes

**0**answers

90 views

### On a variant of Eidelheit's theorem

A theorem of Eidelheit from 1940's asserts that two Banach spaces $X$ and $Y$ are isomorphic if and only if $L(X)$ and $L(Y)$, the algebras of all bounded linear operators, are isomorphic as Banach ...

**0**

votes

**0**answers

60 views

### Can sum of two (co)type $p$ subspaces fail to have the same (co)type?

Type and cotype are very pleasant invariants of Banach spaces. However answer to the following question seems to be missing from the literature.
Let $X$ be a Banach space and suppose that $Y,Z$ are ...

**8**

votes

**2**answers

222 views

### ODE properties true in finite dimension but not in Banach spaces of infinite dimension

Some properties of Ordinary Differential Equations - ODE are true in finite dimension spaces but not in Banach spaces of infinite dimension.
The first one I know is the Peano existence theorem. I ...

**1**

vote

**0**answers

104 views

### The category of discontinuous Banach spaces

A banach space is discontinuous if it is isometric to $DC(X)$ for some Hausdorff topological space $X$. ($DC(X)$ is defined here. We denote by $DBan$, the category of all discontinuous ...

**3**

votes

**0**answers

112 views

### Norm condition in a Banach lattice

Consider the following "condition (J)" on the norm of a (real or complex) Banach lattice $E$: whenever $x$ and $y$ are disjoint (i.e., $|x|\wedge |y|=0$) then
$\|x+y\|+\|x-y\|=2\|x\|+2\|y\|$.
...

**13**

votes

**0**answers

274 views

### $C^*$-algebra generated by those operators that are bounded on every $\ell_p$

Suppose $T: c_{00} \to c_{00}$ is a linear map such that, when regarded as an infinite matrix, there is a uniform bound on the $\ell_1$-norms of its columns, and a uniform bound on the $\ell_1$-norms ...

**0**

votes

**0**answers

237 views

### Banach space of discontinuous functions on a product space

Edit: According to comments of Eric Wofsy and Yemon Choi I edit the question.
For a (compact) topological space $X$, we put $A=\{f:X\to \mathbb{C}\mid f\text{ is bounded}\}$. We define a ...

**4**

votes

**0**answers

184 views

### Dual or pre-dual of BV

Was there any relevant work to determine the dual (or more likely the predual) of the space of bounded variation functions $BV(\mathbb{R}^n)$ (I recall the definition : a function in ...

**0**

votes

**2**answers

110 views

### Book and Papers for properties of uniformly convex and locally uniformly convex and strictly convex Banach spaces.

I am looking for reference books and research articles which cover analysis of uniformly convex and locally uniformly convex and strictly convex Banach spaces.