**2**

votes

**1**answer

155 views

### Nuclear vs Integral operators on Hilbert spaces

Consider the Hilbert space $L_2=L_2[0,1]$. Is it true that for each nuclear (trace-class) operator on $L_2$ there exists a function $K\in L_1(L_2)$ such that
$$Tf = \int\limits_0^1 K(s) f(s) ...

**2**

votes

**0**answers

180 views

### Banach space interpolation theory in terms of categories

I have recently learned a bit about higher category theory.
And there is a question I would like to ask. Can we enrich banach space interpolation theory by using higher category theory?
Is it ...

**5**

votes

**3**answers

342 views

### Structure of an intersection of $L^p$-spaces

In what follows, $L^p$ denotes the space of functions from $\mathbb{R}$ to $\mathbb{R}$ such that $\int_{\mathbb{R}} |f(x)|^p\mathrm{d}x < \infty$.
I am interested to understand the structure we ...

**6**

votes

**0**answers

455 views

### Products of spaces containing no copies of $\ell_2(\Gamma)$

Given an infinite set $\Gamma$, I would like to know if the class of Banach spaces containing no copies of $\ell_2(\Gamma)$ is stable under finite products.
When $\Gamma$ is countable the answer is ...

**1**

vote

**0**answers

61 views

### Is $\partial^\alpha$ a map $H^{s,p}(\mathbb R^N,F)\to H^{s-|\alpha|,p}(\mathbb R^N,F)$?

More precisely, the question is formulated as follows. Let $F$ be an arbitrary Banach space, especially not having the UMD−property. Let $N\in\mathbb N$ and $s\in\mathbb R$ and $1\le p\le +\infty$ . ...

**0**

votes

**0**answers

76 views

### What's the dual space of $c_{0}^{\mathcal {A}}(X)$?

Suppose that $X$ is a Banach space and $({\mathcal {A}},\alpha)$ is a Banach operator ideal. A sequence $(x_{n})_{n=1}^{\infty}$ in $X$ is said to be ${\mathcal {A}}-$convergent to zero if there exist ...

**1**

vote

**1**answer

180 views

### Final topology of surjective linear map on Banach space

Let $L:X\rightarrow Y$ be a surjective linear map from Banach space $(X,||\cdot||_X)$ to vector space $Y$ and denote with $\tau_L$ the final topology on $Y$ induced by $T$.
Is $\tau_L$ equivalent ...

**4**

votes

**0**answers

144 views

### Ultrapowers and complemented subspaces

Let $Y$ be a closed subspace of a Banach space $X$, and let $\mathcal{U}$ be a nontrivial ultrafilter on the set $\mathbb{N}$ of all integer numbers.
It is not difficult to see that if $Y$ is ...

**5**

votes

**1**answer

173 views

### Introducing a dual space structure

Suppose that we have a Banach space $X$ together with some locally convex Hausdorff topology on $X$, weaker than the one given by the norm, which makes the unit ball of $X$ compact. Is $X$ ...

**3**

votes

**1**answer

172 views

### A differentiable version of the Michael selection theorem

Assume that $X$ and $Y$ are Banach spaces and $T:X\to Y$ is a bounded surjective linear map.
Is there a Gateaux differentiable function $g:Y\to X$ such that $T\circ g=Id_{Y}$?

**1**

vote

**0**answers

97 views

### Mixed Tsirelson Norm

A couple of days ago I posted this question on Mathematics Stack Exchange. Surprisingly, so far, I haven't received any answers or comments about it (besides my own possible answer). Maybe I can get ...

**3**

votes

**4**answers

453 views

### Understanding reasons for best constants in inequalities

Why, in functional analysis, is so important to calculate best constant in an embedding inequality?
Cross-posted from ...

**3**

votes

**1**answer

254 views

### A (non trivial) continuous map on a Banach space which is nowhere Frechet differentiable

Assume that $X$ is a Banach space. Is there a continuous map $f:X\to X$ such that $f$ is nowhere Frechet differentiable, but its restriction to every finite dimensional subspace is every where Frechet ...

**3**

votes

**1**answer

330 views

### A useful criterion in vector integration

I would like to know the proof of the following theorem:
Consider two Banach spaces $X\hookrightarrow Y$ and $1<p,q\leq\infty$. Let $(f_n)_{n\geq 0}$ be a bounded sequence in $L^q(I,Y)$ and let ...

**5**

votes

**1**answer

170 views

### Complemented subspaces of ultrapowers

It's a famous result of Maurey that a Banach space $E$ is finitely representable in $X$ if and only if it is a subspace of some ultrapower of $X$. Is there an analogous result for complemented ...

**5**

votes

**1**answer

204 views

### Banach spaces with no reflexive complemented subspaces

If $X$ is a Banach space with the Dunford Pettis Property (DPP), then no infinite reflexive subspace can be complemented. Suppose now that the Banach space has the property, that no infinite reflexive ...

**0**

votes

**1**answer

68 views

### Non-strictly singular quotients

Every separable subspace, in particular, $\ell_p$ for $p\in (1,\infty)$ is a quotient of $\ell_1$. However, every map from $\ell_1$ to $\ell_p$ is strictly singular (as $\ell_1$ is self-saturated). ...

**3**

votes

**1**answer

248 views

### Separable Banach spaces which are absolute Lipschitz retracts

A subset $F$ of a metric space $M$ is called a Lipschitz retract of $M$ if there is a Lipschitz map from $M$ onto $F$ which coincides with the identity on $F$. A metric space which is a Lipschitz ...

**0**

votes

**0**answers

104 views

### On the existence of embeddings of $\ell_r$ into $L_1([0,1], \ell_p)$ for $r<p$

If $2<r<p$, is it true or false that $\ell_r \not \!\hookrightarrow L_1([0,1], \ell_p)$ ? In other words, if $r< p$, is it true or false that $L_1([0,1], \ell_p)$ contains a copy of $\ell_r$ ...

**2**

votes

**0**answers

86 views

### Birkhoff orthogonal of a Banach space in its bidual

Let $X$ be a Banach space embedded in $X^{**}$ in the usual way.
We consider the set
$$
O_X := \{ x^{**}\in X^{**} : \|x^{**}-x\|\geq \|x\| \textrm{ for all }x\in X\}.
$$
I think this is the ...

**7**

votes

**1**answer

382 views

### Existence of injective operators with dense range

Given two separable (infinite dimensional) Banach spaces $X$ and $Y$, it is not difficult to show that there exists an injective (bounded linear) operator $T:X\to Y$ with range dense in $Y$. See S. ...

**1**

vote

**1**answer

106 views

### Contractively complemented subspaces without contractively complemented complement

Can someone give me an example of a Banach space $X$ and contractive projection $P\in\mathcal{B}(X)$ such that $\ker P$ is not a range of any contractive projection $Q\in\mathcal{B}(X)$?

**6**

votes

**1**answer

269 views

### Morita equivalence for operator algebras and tensor products, question about proof

This is a bit of a dumb question I know, but I was reading "Morita equivalence for C*-algebras and W*-algebras" by Rieffel, in this section about Morita equivalences and how they relate to forming ...

**2**

votes

**1**answer

185 views

### tensorial product with Lp

Let us consider E a finite-dimensional normed space on $\mathbb{R}$ and a real number $p\geq 1$.
Is it true that the projective tensorial product $E \widehat{\otimes}_\pi L^p(\mathbb{R},\mathbb{R})$ ...

**3**

votes

**1**answer

242 views

### When is the bound in Riesz-Thorin Interpolation Theorem attained?

Let me recall the statement of Riesz-Thorin theorem (see also http://en.wikipedia.org/wiki/Riesz%E2%80%93Thorin_theorem).
Theorem (Riesz-Thorin): Let $(X,\mu)$ and $(Y,\nu)$ be $\sigma$-finite ...

**4**

votes

**2**answers

262 views

### Banach-Mazur distance to complex $\ell^1$ of a space containing real $\ell^1$

Consider a complex Banach space $X$ with a real subspace isometric to $\ell^1_{\mathbb R}$. What is the best constant $c$ such that $X$ contains a complex subspace $c$-isometric to $\ell^1_{\mathbb ...

**2**

votes

**0**answers

173 views

### Centralizers and containment of $c_0$

I have this question also in MSE (see: http://math.stackexchange.com/questions/666053/centralizers-and-containment-of-c-0), but I have not got an answer there. So I thought I try my luck here.
Let ...

**2**

votes

**1**answer

174 views

### On sequences which converge to zero with respect to an operator ideal

Let $X$ be a Banach space and $\mathcal{A}$ be an operator ideal. A sequence $(x_{n})_{n=1}^{\infty}$ in $X$ is called $\mathcal{A}$-convergent to zero if there exist an operator $S\in ...

**5**

votes

**0**answers

217 views

### On the classification of injective Banach spaces

A Banach space $E$ is injective when it is complemented in each Banach space $X$ that contains it as a closed subspace. The space $E$ is $1$-injective if the copy of $E$ in $X$ is the range of a ...

**7**

votes

**3**answers

329 views

### Rosenthal like inequality for weak $\mathbb L^p$-norms

Let $p$ be a real number greater than $1$. It is well known (see Hall and Heyde's Martingale limit theory and its applications, Theorem 2.10) that there exists a constant $C_p$ such that if ...

**0**

votes

**1**answer

270 views

### Constructing a continuous matrix valued function

Given $d<k$. Let ${\cal M}_{d\times k}(\mathbb{R})$ denotes the set of all $d\times k$ real matrices and suppose that $H:\mathbb{R}^k\rightarrow {\cal M}_{d\times k}(\mathbb{R})$ is a continuous ...

**9**

votes

**1**answer

260 views

### Grothendieck spaces and total subspaces of the dual

There is probably an embarrassingly simple counter-example to my question but I couldn't figure it out myself. Let me give it a try here.
A Banach space $X$ is Grothendieck if weak*-convergent ...

**0**

votes

**3**answers

223 views

### The importance of basis constant in Banach spaces

Let $X$ be a Banach space an let $(e_n)_{n=1}^{\infty}$ be a Schauder basis of $X$. If we denote the sequence of the natural projections associated with $(e_n)_{n=1}^{\infty}$ by ...

**4**

votes

**1**answer

171 views

### A homogeneous but slightly asymmetric inequality

I need to prove the following inequality: for any $Z=(z_1,\dots,z_l)\in\mathbb{C}^l$ for any $p\geq 2$ and $l\geq 2$
\begin{equation}
\left|\left|\sum_{j=1}^l ...

**0**

votes

**1**answer

116 views

### Commutativity of convex hulls and closed balls

Let $X$ be a closed convex subset of a Banach space, and let $A\subseteq X$ be a Borel set. Denote
$$
B_r(A) :=\{y\in X:\exists x\in A \text{ such that }\|x-y\|\leq r\}
$$
and by $H(A)$ the convex ...

**1**

vote

**2**answers

167 views

### Self-Adjointness for Banach Spaces

Good evening. Is there a reasonable notion of being self-adjoint for the adjoint operator on Banach Spaces? Kind regards, Alex

**2**

votes

**2**answers

106 views

### Representation of Banach spaces partially ordered by solid, normal, minihedral cones

I've been using the representation result below, from Krasnosel'skij/Lifshits/Sobolev; Positive Linear Systems---The Method of Positive Linear Operators. Heldermann Verlag, 1989.
Theorem. Let $E$ be ...

**11**

votes

**1**answer

229 views

### Containment of $c_0$

I have the following question. I guess it's quite simple for experts.
Unfortunately, I could not come up with an answer yet.
Let $X$ be a Banach space which contains no copy of $c_0$.
Does it impply ...

**14**

votes

**0**answers

458 views

### An open problem in convex geometry

Is it possible to find four norms $\| \cdot\|_k$ $( 1 \leq k \leq 4)$ on the plane such that a three-dimensional normed space containing four subspaces isometric to these normed planes does not exist? ...

**1**

vote

**0**answers

67 views

### Suprema and infima in spaces ordered by non-normal cones

Background
We shall call a subset $V_+ \subseteq V$ of a Banach space $V$ a cone if
$V_+$ is closed,
$\alpha V_+ \subseteq V_+$ for all $\alpha \geqslant 0$, and
$V_+ \cap (-V_+) = \{0\}$.
Cones ...

**1**

vote

**0**answers

87 views

### two concepts of positivity for elements of $C(X)$ when $X$ is hyper-stonean

Suppose that $X$ is a compact space. Let $M(X)=C(X)^*$ denote the Banach space of regular measures. Is the following statement true:
$F:M(X)\to\mathbb{C}$ is a positive functional if and only if the ...

**3**

votes

**2**answers

192 views

### Is the space of trace class operators finitely representable in an $L^1$-space?

I am interested in knowing whether the space of trace class operators is (crudely) finitely representable in an $L^1$-space. I suspect that the answer is negative but I am unable to find any argument ...

**3**

votes

**2**answers

210 views

### Banach algebra for measures induced by Haar measures

It is classical that $L^1(G, m)$ is a Banach algebra when $G$ is a locally compact group with Haar measure $m$ by using the operation of convolution via the integral
...

**5**

votes

**1**answer

169 views

### An extreme point of the ball of the space of compact operators

It is very easy to see that the unit ball of $c_0$ has no extreme points. I was trying to spot any extreme points in the unit ball of the space of compact operators on a Hilbert space (a ...

**4**

votes

**0**answers

154 views

### How to check a function belongs to a Banach space

My question comes from my confusion in studying the Fefferman-Stein inequality which says that for any $f\in L^p$ it holds $\|f\|_p\leq c \|f^\#\|_p$ with $f^\#$ the maximal function. This is an ...

**6**

votes

**0**answers

352 views

### Second duals of Grothendieck spaces

The classical example of a Grothendieck space is $\ell_\infty$. It is also known that its even duals $\ell_\infty^{**}$, $\ell_\infty^{(4)}$, $\dots$ are Grothendieck spaces.
(See, e.g., this note ...

**2**

votes

**1**answer

247 views

### Is any order bounded continuous linear functionals a difference of positive continuous functionals?

Let $B$ be a Banach space and $K$ a closed proper cone in $B$ such that the induced partial order makes $B$ a vector lattice. Let $K'=\{x^*\in B':\langle x^*, x\rangle\geq 0\ \forall x\in K\}$ the ...

**1**

vote

**1**answer

581 views

### Precise versions of “differential operators are unbounded but closed linear operators”

I am trying to understand to what extent the following result of Hille is an extension of the usual theorems on differentiation under the integral sign.
Theorem (Hille). Let $(\Omega,\Sigma,\mu)$ ...

**12**

votes

**1**answer

360 views

### Quotients of $\ell_\infty$ by separable subspaces

Given a (closed) separable subspace $M$ of $\ell_\infty$, I am interested in conditions implying that the quotient $\ell_\infty/M$ is isomorphic to a subspace of $\ell_\infty$.
It is not difficult ...

**7**

votes

**1**answer

130 views

### Quasi-reflexive spaces which are not isometric to dual spaces

My question may sound weird and I have no deep motivation behind it other than curiosity.
As is well-known, quasi-reflexive spaces have the Radon-Nikodym property hence their balls have lots of ...