3
votes
2answers
144 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 ...
5
votes
0answers
159 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 ...
0
votes
0answers
38 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 $1\le p\le +\infty$ . ...
1
vote
1answer
154 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 ...
3
votes
0answers
116 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
1answer
151 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
1answer
134 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
0answers
83 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
1answer
202 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
1answer
285 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
1answer
160 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
1answer
158 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 ...
3
votes
1answer
166 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
0answers
121 views

On ultrafilters

Let $\mathcal{U}$ be a free ultrafilter of $\mathbb{N}$. Is true or false what $(C([0,1],\ell_p)^*)_{\mathcal{U}} = L_1(\mu, \ell_q)$, where $\mu$ is a measure, $C([0,1],\ell_p)^*$ is the dual of ...
0
votes
0answers
90 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
0answers
77 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 ...
6
votes
0answers
184 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
1answer
86 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 $Q\in\mathcal{B}(X)$?
3
votes
1answer
164 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
1answer
153 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
2answers
226 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
0answers
157 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 ...
1
vote
1answer
159 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 ...
4
votes
0answers
155 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 ...
0
votes
0answers
60 views

Open ideally convex sets

Background Recall that a subset $A \subseteq X$ of a Banach$^{1}$ space $X$ is said to be ideally convex if, for every bounded sequence $(x_n)_{n \in {\mathbb N}}$ in $A$ and every sequence ...
7
votes
3answers
274 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 ...
8
votes
1answer
221 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
3answers
168 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 ...
2
votes
2answers
97 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
1answer
186 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 ...
1
vote
0answers
55 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 ...
3
votes
1answer
122 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
2answers
189 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 ...
2
votes
1answer
182 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
1answer
349 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 $(\Omega,\Sigma,\mu)$ ...
11
votes
1answer
271 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
1answer
108 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 ...
14
votes
1answer
409 views

Meager subspaces of a Banach space and weak-* convergence

I previously asked a version of this question on Math.SE, but didn't receive an answer. (But there is a bounty there if you want to claim it!) Let $X$ be a Banach space. (If it helps, feel free to ...
6
votes
0answers
119 views

Complex interpolation of a Banach space and its antidual when the space has a basis

Given a complex Banach space $X$ and its antidual $\hat X^*$, it is possible in some cases to apply the complex interpolation method, and get as $(X,\hat X^*)_{1/2}$ a Hilbert space. See [F. Watbled, ...
6
votes
1answer
209 views

How hard (P, NP, NP-hard) is it to compute Schur norms of matrices (as multipliers)?

Given a matrix $A\in M_n(\mathbb{C})$, I will denote by $||A||_\infty$ the operator norm of $A$, as seen acting on the Hilbert space $\mathbb{C}^n$. This makes $M_n(\mathbb{C})$ into a Banach space ...
7
votes
3answers
305 views

Dual Banach space of $B(X,Y)$ when $X$ is finite dimensional

Denote $B(X,Y)$ the Banach space of bounded operators between Banach spaces $X$ and $Y$. When $X$ and $Y$ are both finite dimensional, it follows from the formula $$\|u\|_{B(X,Y)} = \sup_{\|x\|_X ...
1
vote
0answers
131 views

Elements of Minimal Norm on an affine subset of a Banach Space

Let $v_1$ and $v_2$ be two elements of a real Banach space $(X,\lVert\;\rVert)$, each of unit norm. Consider the one-dimensional affine subspace $v_1+tv_2$ for real $t$. Is there a general formula for ...
7
votes
1answer
343 views

Non continuous Linear form on $E=C([0,1],\mathbb{R})$ without AC

Let's note $E=C([0,1],\mathbb{R})$ the Banach space of real continuous funtions from the [0,1] interval with the uniform norm. Is it possible to show a non-continuous linear form on $E$ exists ...
0
votes
0answers
149 views

Banach-Mazur distance estimate finite-dimensional $\ell_p$ spaces

Hey my fellow Banach space guys. Sorry for another elementary question, and yes I have looked for the past hour and a half to see if I can find it on Google or in my books at home. Fix ...
2
votes
2answers
171 views

When $L^\infty$ is 1-injective

It is known that when $\mu$ is $\sigma$-finte measure, then $L^\infty(\mu)$ is $1$-injective. But I want to know whether it is right for any $L^\infty$ spaces.
8
votes
0answers
225 views

Ultrapowers of Banach spaces without the continuum hypothesis

Let $\mathcal{U}$ be a non-trivial ultrafilter on the set of integers $\mathbb{N}$, and let $C(K)$ denote the Banach space of continuous functions on a compact $K$. Under the continuum hypothesis CH, ...
2
votes
0answers
98 views

Projective tensor powers of Banach spaces over a normed field

Let $E$ be a Banach space over a complete normed field $\mathbb K$. Is it possible to classify all proper $E$ for which the projective seminorm $p_n$ defined on the $n$-th tensor power of $E$ is a ...
10
votes
2answers
297 views

Pull-back of generalized functions

Let $f\colon X\to Y$ be a smooth map between smooth manifolds. Then the pull-back operation $f^*\colon C^\infty(Y)\to C^\infty(X)$ is a linear continuous operator when $C^\infty$ is equipped with the ...
7
votes
1answer
146 views

Sequential continuity of linear operators

Let $u\colon L\to M$ be a linear map of locally convex linear topological vector spaces. Assume that $u$ is sequentually continuous, i.e. maps convergent sequences to convergent ones. (This notion is ...
2
votes
0answers
100 views

Is reflexive Banach space valued scalarwise Lebesgue space isomorphic to the Bochner space?

I first specify the setting and then formulate the question precisely. (A very long post follows.) Definitions 1. For $E$ a (real Hausdorff) locally convex space, say that $E$ is suitable iff there ...