All Questions
1,222 questions
2
votes
1
answer
578
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 ...
- 717
6
votes
1
answer
548
views
Non-super reflexive space
Suppose $X$ is a reflexive space (possibly non-separable) which is not super-reflexive. Then (by definition) there exists a non-reflexive Banach space $Y$ which is non-reflexive but is finitely ...
- 387
2
votes
1
answer
327
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 ...
- 552
3
votes
2
answers
423
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
$$(f*g)(y)=\int_Xf(x)g(yx^{-1})\,...
- 1,049
4
votes
1
answer
287
views
Second conjugate operators to operators on $c_0$
I posted my question at MS but unfortunately it is still without a response, so let me ask it here.
We can think about a bounded operator $T\colon c_0\to c_0$ as a double-infinite matrix $[T_{mn}]_{m,...
- 51
3
votes
2
answers
599
views
Weak topology of WOT
Let $E$ be a reflexive Banach space and let $B(E)$ be the space of bounded operators on $E$ endowed with the weak operator topology. In particular, the unit ball of $B(E)$ is then WOT-compact. $(B(E), ...
- 115
11
votes
1
answer
504
views
Do ultrapowers of classical Banach spaces have unconditional bases?
I am trying to imagine (to some extent, of course) the geometry of ultrapowers of certain 'easy-to-handle' Banach spaces. Let me start with $X = \ell_p$, $p\in (1,\infty)$ or $X=c_0$.
Since the ...
- 125
8
votes
0
answers
1k
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 norm-...
- 4,461
7
votes
0
answers
559
views
The Banach space of bounded functions with countable support
Let $X$ be a set of cardinality $\aleph_1$ and consider the Banach space $\ell_\infty^c(X)$ of all scalar-valued bounded functions on $X$ which are non-zero only for countably many elements of $X$ ...
- 11.3k
8
votes
0
answers
421
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 finite-...
- 6,206
0
votes
1
answer
128
views
On a theorem by Mooney and Khavin on the weak sequential completeness of the predual of $H^\infty(\mathbb{D})$
There is a theorem by Mooney http://msp.org/pjm/1972/43-2/pjm-v43-n2-p.pdf#page=185 and independently proved by Havin which says that the predual of $H^{\infty}(\mathbb{D}),$
$L^{1}/H^{1}_{0}$ is ...
- 175
4
votes
1
answer
280
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}- \frac{q(t)q((s-...
- 4,705
6
votes
2
answers
1k
views
Closed, complemented subspaces of $l^1(X)$ when $X$ is uncountable
... are all isomorphic to $l^1$ on some other index set. At least, that much I "know" from 2nd-hand sources, since the original proof is apparently in a paper of Köthe from the 1930s 1960s (in ...
- 25.8k
2
votes
1
answer
294
views
Necessary conditions for optimality in Banach spaces
Let $X$ denote the non-negative "orthant" of the Banach space $L^2$ (or whatever you call the set of functions in $L^2$ that are non-negative), and let $C$ be a closed, convex subset of $X$. Let $f$ ...
- 21
0
votes
1
answer
226
views
Subspaces of $H^{\infty}(\mathbb{D})$ which contains a nontrivial weak* closed subalgebra
Let $H^{\infty}(\mathbb{D})$ denotes the Banach space of bounded holomorphic functions in the unit disc. Consider the weak* topology on $L^{\infty}(\mathbb{T})$
that it inherits as the dual of $L^{1}(\...
- 175
2
votes
1
answer
232
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.
...
- 356
0
votes
0
answers
302
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 semi-...
- 356
6
votes
0
answers
365
views
Approximating a measurable function from a second-countable, locally compact Hausdorff group to a separable Banach space
Let $ G $ be a second-countable, locally compact Hausdorff group and $ B $ a separable Banach space.
We say that a function $ f: G \to B $ is Bochner-measurable if and only if it is the everywhere ...
- 1,396
5
votes
1
answer
508
views
Projections which are not completely bounded
There are 'canonical' examples of maps on operator spaces which are not completely bounded. Nevertheless, I couldn't produce any examples of bounded projections on relatively easy to understand ...
- 181
4
votes
1
answer
207
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 $X$ (Banach-...
- 11.3k
6
votes
0
answers
532
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 ...
- 4,461
3
votes
1
answer
365
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 ...
- 315
7
votes
0
answers
183
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 ...
- 71
4
votes
1
answer
419
views
Pitt's theorem for non-separable $\ell_p$ spaces
A short variant of Pitt's theorem is the followig: for $1\leq p < r <\infty$ holds
$$
\mathcal{B}(\ell_r(\mathbb{N}),\ell_p(\mathbb{N}))=\mathcal{K}(\ell_r(\mathbb{N}),\ell_p(\mathbb{N}))
$$
Now ...
- 1,697
1
vote
1
answer
251
views
James $\ell_1$-theorem
This question was asked at MSe before but with no answer.
I am struggling with the very last estimate in the proof of James' $\ell_1$-theorem. (Please see below an excerpt from Albiac and Kalton's ...
- 13
7
votes
0
answers
708
views
Minkowski's Inequality for Integrals in Orlicz spaces
EDIT: I have changed the question to have less parameters, fitting it into the context of Orlicz spaces.
Suppose $f:[0,\infty)\to[0,\infty)$ is convex and increasing, $f^{-1}:[0,\infty)\to[0,\infty)$...
- 882
4
votes
0
answers
114
views
Weakenings of the Bounded Approximation Property
Let $X$ be a Banach space, and let $F(X)$ be the finite-rank operators on $X$.
Then $X$ has the $\lambda$-BAP if there exists a net $(S_i)_i$ in $F(X)$ with $\sup_i\|S_i\|\leq\lambda$ such that $...
- 3,497
7
votes
2
answers
484
views
Extension of weakly compact operators from $\ell_1$ into $c_0$
Is every weakly compact operator from $\ell_1$ into $c_0$ extendible
to any larger space? Equivalently, is every weakly compact operator from $\ell_1$ into $c_0$ extendible to $\ell_\infty$?
1
vote
1
answer
237
views
Interpolation and embeddings for parabolic function spaces
I have a somewhat easy looking question on parabolic function spaces:
Let $B$ be a ball in $\mathbb R^n$ and let $T>0$. Denote $Q:=B \times [0,T]$. Assume $f \in L^2(Q) \cap L^\infty(0,T; L^q(B))$ ...
- 632
2
votes
1
answer
373
views
Is it true that $c_0(X)^* = \ell_1(X^*)$ ?
I'm trying to prove this that but I can't . Any help/reference ?
- 151
7
votes
2
answers
808
views
Is a subspace with a certain property dense in the dual of a vector space?
Suppose we have a normed vector space $V$ and its dual $V^*$, and suppose that $X \subseteq V^*$ has the property that for every $v \in V$, there is some $\phi \in X$ with $\Vert \phi \Vert = 1$ such ...
- 73
2
votes
2
answers
214
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 ...
- 123
3
votes
1
answer
255
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 $T\circ g=Id_{Y}$?
- 356
4
votes
1
answer
615
views
Isometric embeddings of $\ell_q^m$ into $\ell_p$ and $L_p$ for $p,q\in[1,+\infty]$
I'm looking for articles describing or proving nonexistence of isometric embeddings of $m$-dimensional space $\ell_q^m$ into $L_p$ and $\ell_p$ for $q,p\in[1,+\infty]$.
Since $\ell_q^m$ is finite ...
- 1,697
4
votes
0
answers
242
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 $B$, $\...
- 4,679
1
vote
1
answer
160
views
Transferring locally uniformly convex norm by bounded linear operator from one Banach space to another
I'm trying to find a simple proof this theorem
Let $Y$ be a locally uniformly convex Banach space and $X$ be a Banach space and let $T$ be a bunded linear operator from $X$ into $Y$ such that for ...
- 823
2
votes
1
answer
365
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 \mathcal{A}(Z,X)...
- 3,343
7
votes
1
answer
423
views
Best constant in comparison between Rademacher and gaussian averages?
Let $(g_k)$ be a sequence of independent standard gaussians variables on a fixed probability space $\Omega$. Let $(\epsilon_k)$ be a sequence of independent rademacher variables.
What is the best ...
- 1,222
2
votes
1
answer
188
views
Existence of normal structure in strictly convex Banach spaces
Does there exists a strictly convex Banach space which is not uniformly convex and has normal structure ?
- 41
0
votes
2
answers
291
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.
- 823
4
votes
1
answer
902
views
Separable $L_1$-predual
Some isometric preduals of $\ell_1$ are of the form $C_0(K)$ where $K$ is countable. I am wondering whether this is a general rule.
Question: Is there a measure $\mu$ and a (preferably separable) ...
- 191
4
votes
2
answers
484
views
When is a metric space isometrically embeddable into some Banach space?
EDIT
Oops---I found the answer to the first question of mine here on Wikipedia---this is really classic material. I'll leave the question open for a bit, in case someone tells me something ...
- 28.6k
2
votes
1
answer
386
views
Decomposing bilinear forms in Hilbert spaces
You are given a complex Hilbert space $H$ with two equivalent Hilbert space structures $<,>$ and $<,>'$. Define $<,>''=<,> + <,>'$ to be the sum of our two scalar ...
- 1,211
0
votes
1
answer
861
views
Norms agreeing on dense subspace [closed]
Suppose $(B,\|\cdot\|)$ is a Banach space, $V\subset B$ a dense subspace, and $V$ is equipped with a norm $\|\cdot\|_V$ such that $\|x\|_V = \|x\|$ for all $x\in V$.
Is $(B,\|\cdot\|)$ a completion ...
- 143
1
vote
1
answer
353
views
Agreement of two topologies on a linear space
I'm dealing with the formalism of an abstract Wiener space, and I'm not sure if two relevant topologies coincide.
Let $X$ be a topological vector space, and let $X^*$ be its dual space of continuous ...
- 8,512
3
votes
1
answer
245
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.
...
- 4,432
6
votes
0
answers
161
views
Real interpolation space between the Wiener algebra and $L^2$
The Wiener algebra $W_n$ is the image by the Fourier transform of $L^1(\mathbb R^n)$. What is the (complex) interpolation space between $W_n$ and $L^2(\mathbb R^n)$? It is probably not true that for $\...
- 16.2k
2
votes
0
answers
648
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 $\...
- 4,907
3
votes
2
answers
362
views
Invariant subspaces for compact restrictions
Suppose $Y$ is a closed hyperplane in $X$, so we can write $X=Y\oplus[x_0]$. Let $y_n$ be a normalized basis of $Y$. Define an operator $S:Y\to Y\oplus[x_0]$ by $Sy_n=\alpha_ny_n+\beta_nx_0$, for any $...
- 31
4
votes
2
answers
676
views
Basis for L_infty(R)
Let $V$ be the Banach space of bounded sequences of reals with the sup norm. Does there exists a subset $B$ of $V$ such that
Linear Independence: For all functions $c$ in $\mathbb{R}^B$, if $\sum_{b ...
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