A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

learn more… | top users | synonyms

5
votes
0answers
198 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
65 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
315 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
1answer
207 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
1answer
250 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
208 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
1answer
164 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
1answer
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
2answers
160 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
2answers
103 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
217 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
0answers
434 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
0answers
66 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
0answers
82 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
2answers
181 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
204 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
1answer
164 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
0answers
149 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 ...
5
votes
0answers
211 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., ...
2
votes
1answer
226 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
489 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
1answer
343 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
127 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 ...
16
votes
1answer
493 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 ...
4
votes
1answer
159 views

When does $\ell_1(\Gamma)$ embed into $L_1(\mu)$?

Suppose we are given an uncountable set $\Gamma$ and a measure space $(\Omega, \mathcal{F}, \mu)$. I would like to know when the Banach space $\ell_1(\Gamma)$ embeds into $L_1(\mu)$. Of course, this ...
6
votes
0answers
134 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, ...
10
votes
2answers
457 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
350 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 ...
4
votes
0answers
131 views

Is the space $U_{0}$ constructed by J.Lindenstrauss Lipschitz complemented in $l^{1}$?

In $l^{1}$, we let $x_{n}=e_{n}-\frac{e_{2n+1}+e_{2n+2}}{2}(n=1,2,...)$, where $(e_{n})_{n=1}^{\infty}$ is the unit vector basis in $l^{1}$. We let $U_{0}$ be the subspace generated by the monotone ...
1
vote
1answer
90 views

Banach Isomomorphic Cts Fucntion Algebras for two Non-Homeomorphic Top Spaces?

Can anyone provide me with an example of two non-homeomorphic locally-compact Hausdorff spaces $X$ and $Y$, such that $C(X)$ and $C(Y)$ are isomorphic as Banach algebras. Clearly, the ...
1
vote
0answers
156 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
385 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
170 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
1answer
176 views

Examples of non super-reflexive spaces

What are good examples of Banach spaces which are and aren't super-reflexive? Whenever properties of Banach spaces like super-reflexivity, uniform convexity etc. are discussed, my impression is that ...
3
votes
2answers
253 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
244 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
105 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 ...
9
votes
2answers
364 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 ...
6
votes
1answer
171 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 ...
1
vote
0answers
123 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 ...
0
votes
0answers
181 views

Closed-form expressions for dual norms of real normed vector spaces

Didn't get any biters over at MSE, so I figure this place might be more appropriate... Say that $V$ is a finite-dimensional real normed vector space, where for some $v \in V$ the norm is notated by ...
1
vote
0answers
72 views

Maximum Principle with Banach Control Space

This is a problem that seems very natural to me, but I couldn't find any formal statement in the literature for some time now. I am basically considering an autonomous optimal control problem in ...
3
votes
1answer
209 views

Orthonormal basis in $\ell^n_p$

Given a $k$-dimensional subspace in $\ell^n_p$, is there a way to bound the value of $$ \sum_{i=1}^k \|a_i\|_{\ell^p}^2 $$ for $a_i$ an orthonormal (for the "standard" underlying $\ell^n_2$) basis. ...
1
vote
0answers
85 views

Volume Function on Banach Spaces

I'm looking for a reference for the following so-called Volume Function $V_n$, which is intended to be a Banach/normed vector space generalization of the determinant. Let $X$ be a Banach space with ...
1
vote
0answers
95 views

Extenstions of Urysohn's inequality

A version of Urysohn's inequality states that for a symmetric convex body $K \subset \mathbb{R}^n$, one has $$ \left(\frac{\text{vol}(K)}{\text{vol}(B_2)} \right)^{1/n} \le \frac{1}{\sqrt{n}} E \; \| ...
3
votes
1answer
171 views

Does the topological Varopoulos algebra consist of functions that are continuous and Varopoulos norm bounded?

Let $X_1,\dots,X_n$ be compact Hausdorff spaces. Let's define the Varopoulos algebra as the projective tensor product: $$V(X_1,\dots,X_n) := C(X_1) \hat{\otimes} \dots \hat{\otimes} C(X_n),$$ i.e. the ...
1
vote
0answers
52 views

A question about the $C_\lambda$ condition in Banach spaces [closed]

Let $X$ be a Banach space and $K$ its non-empty subset. Let $S:K\to X$ be a single-valued mapping. Then $S$ satisfies the condition $(C_\lambda)$ if for $x,y\in K$ and $\lambda\in(0,1)$, ...
10
votes
1answer
449 views

Quotients of l^infty

Let $M$ be a closed subspace of $l^\infty$. Suppose that the quotient $l^{\infty}/M$ is isomorphic to $l^\infty$. Is it true that $M$ is complemented in $l^\infty$?
2
votes
0answers
200 views

Cameron Martin space

I have seen two definitions of Cameron Martin space of a Gaussian measure $\nu$ on a Banach space (say $W$) and am unable to establish their equivalence. Any help would be appreciated. 1) It is the ...
4
votes
1answer
227 views

Approximation of an integral over the unit ball of L_1

For every $\varepsilon>0$ find a piecewise continuous function $q:[0,1]\rightarrow \mathbb{R}$ such that $\int_0^1 q(x)dx=1$ and $$\int_{0}^1 \int_{0}^{s} \left|\frac{q(s)q(t/s)}{s}- ...