**1**

vote

**2**answers

203 views

### Geodesic on Banach Manifold [closed]

Is there a way of defining a geodesic on a Banach Manifold $M$ which is not itself a Hilbert Manifold?

**2**

votes

**1**answer

83 views

### Weakly $p$-summable sequences in $L_{r}$

By Bessaga-Pelczynski Selection Principle, it is easy to check that both $l_{p}(1\leq p<2)$ and $l_{r}(1<r<p^{*})$ contains no normalized weakly $p$-summable sequences. I do not know if it is ...

**3**

votes

**1**answer

126 views

### $M_Λ(A) → A ⊗ M_Λ(C)$

I saw this in here. Let $A$ be a Banach algebra, and let $\Lambda$ be a non-empty set. We denote by
$M_\Lambda(A)$ be the set of $\Lambda\times\Lambda$ matrices $(a_{ij})_{i,j\in\Lambda}$ with entries ...

**0**

votes

**0**answers

54 views

### skorokhod integral as Weak Integral

Is it possible to express the skorokhod integral on a Banach space $B$ as a special case of the weak (or Pettis) integral over an appropriate Banach space $E$?
For example if $E$ is the space of ...

**2**

votes

**0**answers

117 views

### Weakly null sequences in $X^{**}/X$

Let $X$ be a space and $Q_{X}:X^{**}\rightarrow X^{**}/X$ be the canonical quotient map. Given a weakly null sequence $(f_{n})_{n}$ in $X^{**}/X$. Is there a weakly Cauchy sequence $(x^{**}_{n})_{n}$ ...

**0**

votes

**2**answers

112 views

### Weak continuity of a vector valued function [closed]

Let $f:[0,1]\to \ell_\infty[0,1]$ be defined by $f(t)=\chi_{[0,t]}$. Is it true that $f$ is weakly continuous almost everywhere w.r.t. Lebesgue measure ? Here $\ell_\infty[0,1]$ represents the ...

**24**

votes

**0**answers

489 views

### Isometric embeddings of finite subsets of $\ell_2$ into infinite-dimensional Banach spaces

Question: Does there exist a finite subset $F$ of $\ell_2$ and an infinite-dimensional Banach space $X$ such that $F$ does not admit an isometric embedding into $X$?
There are some results of the ...

**1**

vote

**1**answer

99 views

### A question about weak convergence on the unit ball of a reflexive space

Which class of reflexive spaces $X$ having the property: if a sequence $(x_{n})_{n}\subset B_{X}$ converges to $x$ weakly and $\|x_{n}\|\rightarrow 1$, then the norm of $x$ must be 1. Of course, the ...

**6**

votes

**1**answer

151 views

### Is every ideal part of an operator ideal?

An operator ideal $\mathfrak J$ is a class of continuous operators. Namely, for every pair of complex Banach spaces, $\mathfrak X,\mathfrak Y$, we have that $\mathfrak J(\mathfrak X,\mathfrak Y) \...

**0**

votes

**1**answer

89 views

### On the separability of operator range

Let $T$ be an operator from a Banach space $X$ into a Banach space $Y$ and $1\leq p<\infty$. If $ST$ is compact for any operator $S$ from $Y$ into $l_{p}$, Is $T(X)$ separable? Or under what ...

**5**

votes

**2**answers

178 views

### Density of Gaussian measures on Banach spaces

I am trying to get my head around this question and was reading (1) which states the same a little bit more general:
Let $X$ be a separable Banach space and $X^*$ the dual space. The mean
value $...

**8**

votes

**0**answers

250 views

### Outline of Generic Separable Banach Spaces don't have a Schauder Basis

So, I know P. Enflo showed that there is a separable Banach Space that doesn't satisfy the approximation property. My professor mentioned during class that in fact generic separable Banach Spaces don'...

**0**

votes

**0**answers

43 views

### On characterizations of $p$-integral operators

In most of papers and classical text books, $p$-integral operators are defined and characterized by operators and commutative diagrams. This is quite different from $p$-summing operators and $p$-...

**0**

votes

**1**answer

69 views

### Embedding Riemmanian Manifold Linearly

Given a Hilbert Manifold $M$ does there exist a smooth map into some very large Hilbert space taking geodesics to straight lines?

**2**

votes

**1**answer

58 views

### A question on unconditionally $p$-summable sequences

We say that a sequence $(x_{n})_{n}$ in a Banach space $X$ is unconditionally $p$-summable ($1\leq p<\infty$) if
$$\sup_{x^{*}\in B_{X^{*}}}(\sum_{n=m}^{\infty}|\langle x^{*},x_{n}\rangle|^{p})^{\...

**3**

votes

**0**answers

66 views

### On the relationship between the factorizations of an operator $T$ and its second adjoint $T^{**}$

Let $T$ be an operator from a space $X$ into a space $Y$ and let $1\leq p<\infty$. If $T^{**}$ has a factorization $T^{**}=RS:X^{**}\xrightarrow{S} l_{p}\xrightarrow{R}Y^{**}$, where $S$ is compact ...

**2**

votes

**1**answer

77 views

### Post composition of integral

Setup:
If $\langle \Omega, \mathfrak{F},\mu \rangle$ is a measure space, $f:\Omega \rightarrow E$ is a weakly-measurable function to a Banach space $E$, $g: E \rightarrow E'$ is a diffeomorphism and ...

**1**

vote

**0**answers

68 views

### On compact operators with domain $c_{0}$

Let $X$ be a Banach space and $T$ be a compact operator from $c_{0}$ into $X^{*}$. Let $\epsilon>0$. Is there a compact operator $S$ from $X$ into $l_{1}$ such that $S^{*}|_{c_{0}}=T$ and $\|S\|\...

**4**

votes

**0**answers

30 views

### How to characterize an operator $T$ that factors through a special space?

Let $T\in \mathcal{L}(X,Y)$ and $1<p<\infty$. My question is: Is there a convienent and useful characterization of the operator $T$ factoring through a space $Z$ satisfying $\mathcal{L}(Z,l_{p})=...

**6**

votes

**1**answer

316 views

### Two vector spaces with homeomorphic open subsets are isomorphic?

Is it true that if $ E,F$ are two topological vector spaces (or say Banach spaces) over $\mathbb{R}$ such that they have nonempty open subsets $U\subset E, V\subset F$ which are homeomorphic, then the ...

**1**

vote

**1**answer

122 views

### An operator factoring through a Banach space containing no copy of $l_{1}$

Is there an operator $T:X\rightarrow Y$ that factors through a Banach space $Z$ containing no complemented copy of $l_{1}$, but does not factor through any Banach space $W$ containg no copy of $l_{1}$?...

**5**

votes

**1**answer

76 views

### A question on characterizing a Banach space containing no copy of $l_{1}$

Let $X$ be a Banach space. My question is: $X$ contains no copy of $l_{1}$ if and only if any operator from $X$ to $l_{1}$ is compact? I guess that the necessary part may be true. But is the ...

**5**

votes

**1**answer

83 views

### A question on compact operators with domain $l_{p}$

Suppose that $T$ is an operator from a Banach space $X$ to a Banach space $Y$. Let $1<p<q<\infty$. If $TS$ is compact for any operator $S:l_{p}\rightarrow X$, is $TR$ compact for any operator ...

**11**

votes

**2**answers

217 views

### Existence of closed operators with arbitrary dense domain of a given banach space

Consider any Banach space $X$, and let $Y$ be any dense subspace, then does it necessarily exist a closed linear operator $T$ defined on $X$, such that the domain of $T$ is exactly $Y$, i.e., $D(T)=Y$?...

**8**

votes

**0**answers

222 views

### What is the name for a Banach space property closed under ultraproducts?

In Banach space theory, a super-property is a property of a Banach space that is preserved under ultrapowers. (Update (2015-09-28): The property must also be closed under isometric embeddings.) (...

**9**

votes

**1**answer

108 views

### Approximation via finite rank Cameron-Martin projections

Let $(W, \|\cdot\|_W)$ be a real separable Banach space equipped with
a non-degenerate Gaussian Borel measure $\mu$. Let $H \subset W$ be
the corresponding Cameron-Martin Hilbert space (also known as ...

**1**

vote

**1**answer

216 views

### does every compact convex set in c0 have but countably many extreme points

This seems plausible, given the properties of the unit ball of $c_0$.
I have a compact set in a complex Banach space $X$ whose closed convex hull has uncountably many extreme points. It would be ...

**1**

vote

**0**answers

145 views

### The dual of the space of smooth functions that vanish at infinity

Let $U \subset \Bbb R ^n$ be an open subset and let $\mathcal C$ be the space of the smooth functions on $U$ that vanish at infinity, endowed with the seminorms $p_\alpha (f) = \sup \limits _{x \in U} ...

**4**

votes

**1**answer

116 views

### How to characterize a Banach space $X$ such that any operator from $X$ to $l_{p}$ is compact?

Let $X$ be a Banach space and $1<p<\infty$. How to characterize $X$ such that any operator from $X$ to $l_{p}$ is compact? Are there any known or new results?

**6**

votes

**1**answer

165 views

### Is $T^{**}$ unconditionally $p$-summing whenever $T$ is unconditionally $p$-summing?

A sequence $(x_{n})_{n}$ in a Banach space $X$ is said to be unconditionally $p$-summable if $$\sup_{x^{*}\in B_{X^{*}}}\Bigl(\sum_{n=m}^{\infty}\lvert\langle x^{*},x_{n}\rangle\rvert^{p}\Bigr)^{1/p}\...

**2**

votes

**0**answers

46 views

### Is any operator from $l_{p}$ to a quotient of $l_{r}$($1\leq r<p<\infty$) compact?

Let $1\leq r<p<\infty$. Let $T$ be an operator from $l_{p}$ to a quotient of $l_{r}$. Is $T$ compact?

**2**

votes

**2**answers

264 views

### “Generalisation” of one-parameter semigroups

Let $(Y,\left\|\cdot\right\|_Y)$ be a Banach space and $A:D(A)\subset Y \to Y$ a closed operator. Studying dynamical systems of the form
\begin{equation}
u'=Au
\end{equation}
quickly leads to the ...

**11**

votes

**1**answer

289 views

### Is there a Gelfand-Naimark-like characterization of group algebras $L_1(G)$?

The Gelfand-Naimark theorem establishes that a complex commutative Banach algebra $A$ with an identity and an involution $x\to x^*$ satisfying $\|x x^*\|=\|x\|^2$ is (isometrically isomorphic to) a $C(...

**3**

votes

**2**answers

239 views

### Is there an easier proof to show that the closed convex hull of a normalized weakly null sequence is weakly compact?

In a paper that I am reading there is a following step:
Let $X$ be a Banach space and let $(x_k) \subset X$ be a normalized sequence that converges weakly to $0$.
Then $\overline{co}(x_k)$ is a ...

**3**

votes

**1**answer

274 views

### Proof of Pinelis (1992) - Banach space inequalities

I am reading Pinelis "An approach to inequalities for the distributions of infinite -dimensional martingales" and cannot follow his proof of Theorem 3:
Let $(f_n)$ be a martingale in a separable ...

**0**

votes

**1**answer

97 views

### Solutions of an nonlinear evolution problem

We consider the following continuous-time nonlinear evolution problem
\begin{equation}
\begin{cases} \dot{y}(t)=Ay(t)+F(y(t),u(t)),\quad t\geq0\\y(0)=f\in\mathcal{X}\end{cases}
\end{equation}
where ...

**0**

votes

**0**answers

88 views

### When do block sequences yield disjoint subspaces?

Let $X$ be a Banach space having a (unconditional, normalized) Schauder basis $(e_n)_n$. Suppose that $Y$ and $Z$ are (closed) block subspaces of $X$ having normalized block bases (with respect to $(...

**2**

votes

**1**answer

184 views

### Completion of $C_0^{\infty}(\mathbb{R}^N)$ with norm $\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $

I have a question that I could not find it any where.
Is the completion of $C_0^{\infty}(\mathbb{R}^N)$ with the respect to norm
$$\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{...

**4**

votes

**2**answers

257 views

### Well-complemented copies of $\ell_p^n$

This must be surely known but I couldn't locate this problem in the literature. It popped out in a priori unrelated approximation problem but if true, would help me greatly.
Let $p\in (1,\infty)$.
...

**5**

votes

**1**answer

181 views

### Is the space of vectorial functions that are Dunford integrable complete?

Let $X$ a Banach Space and $(\Omega, \Sigma, \mu)$ a measure space. A function $F:\Omega\rightarrow X$ is Dunford integrable if $x^\ast\circ F$ is $\mu$-integrable for every $x^\ast\in X^\ast$. The ...

**3**

votes

**2**answers

181 views

### On the Lorentz sequence space $d(w,1)$

I am interested in examples of dual Banach spaces $X$ with the Schur property (weakly convergent sequences in $X$ are norm convergent) like $\ell_1$.
The Lorentz spaces $d(w,1)$ [Lindenstrauss and ...

**6**

votes

**0**answers

131 views

### Is there a quotient of $c_0$ without the approximation property?

The famous example of Enflo of a Banach space without the approximation property is actually a subspace of $c_0$. Is there a quotient of $c_0$ without the approximation property?
This would follow if ...

**7**

votes

**2**answers

311 views

### Non strictly-singular operators and complemented subspaces

If $T$ is a bounded operator which is not strictly singular, acting on a separable Banach space $X$, can one always find an infinite dimensional, closed and complemented, subspace $Y$ such that $T$ ...

**5**

votes

**1**answer

227 views

### Hahn Banach type extension of a Lipschitz map

The problem that I posted was a much generalized form of what I had in my mind. All I want to know the literature of Hahn-Banach type extension of Lipschitz map. I know only about the result by ...

**1**

vote

**1**answer

176 views

### Noncommutative analogs of classical Banach geometric properties

The scale of Schatten-von Neumann classes is noncommutatitve analog of classical $\ell_p$-spaces. A lot of researchers devoted their lives to study Banach geometric structure of these spaces. ...

**4**

votes

**0**answers

130 views

### $L_\infty(\mu)$ spaces non-isomorphic to a dual space

Given a measurable space $(\Omega,\mu)$ such that $L_\infty(\mu)$ is isomorphic to a dual space, $L_\infty(\mu)$ is an injective Banach space. Indeed, given a subspace $Y$ of $X$ and a norm-one ...

**0**

votes

**0**answers

109 views

### Unitarizability of group representations

Let $G$ be a Lie (or more general) group. Consider its continuous representation in a Banach space by isometries, i.e. preserving the Banach norm. Under what conditions this representation is ...

**4**

votes

**2**answers

395 views

### Is a specific sequentially closed subset of $M([0,1])$ closed?

Let $M([0,1])$ be the set of finite signed measures on $[0,1]$
(with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all $\...

**4**

votes

**1**answer

142 views

### A differential equation with continuous coefficient and no solution in a reflexive Banach space?

Is there a reflexive Banach space $B$ and a continuous map $f:B\to B$ such that the differential equation
$$ \frac{d x (t)}{dt} = f(x(t)) $$
with some initial condition $x(0)=x_0$ has no solution?

**-1**

votes

**1**answer

81 views

### Infinitesimal generator is bounded [closed]

Consider a strongly continuous semigroup of bounded linear operators $S(t):X\to X$. The infinitesimal generator of $S(t)$ is the linear operator $A:D(A)\subseteq X \to X$ defined by
$$D(A):=\bigg\...