All Questions
1,222 questions
3
votes
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438
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Separable subspaces in dual spaces
Let $X$ be a Banach space and $Y$ be a separable closed subspace of $X^{*}$. Is there a separable closed subspace $Z$ of $X$ such that $Y$ is isomorphic to a subspace of $Z^{*}$? Thank you!
- 4,713
5
votes
1
answer
242
views
Corson-Lindenstrauss : Weakly compact sets as intersection of finite unions of cells
A theorem of Corson and Lindenstrauss in:
Corson, H. H. and Lindenstrauss, J. “On weakly compact subsets of Banach spaces”. In: Proceedings of the American Mathematical Society 17.2 (1966), pp. 407–...
- 716
0
votes
0
answers
59
views
Restriction to Basis of Cadlag function
If $f \in L^2([0,T])$ then it can be written as
$$
f(t) \triangleq \sum_{i \in \mathbb{N}} c_i e_i(t),
$$
for some sequence $\{c_i\}$ of real numbers and a Schauder basis $\{e_i(t)\}$ of $L^2([0,T])$ ...
- 5,405
3
votes
1
answer
153
views
Example of a strictly cosingular operator whose dual is not strictly singular?
The short version of my question: Suppose $T\in\mathcal{L}(X,Y)$ is strictly cosingular. Must $T^*$ be strictly singular?
The long version.
Let $X$ and $Y$ be Banach spaces, and denote by $\mathcal{...
CommunityBot
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2
votes
1
answer
5k
views
Smooth Approximation of Indicator Function of Convex Sets in $\mathbb{R}^n$
Let $( \mathbb{R}^n, \| \cdot \|_P)$ be the $n$-dimensional Euclidean space equipped with $\ell_p$-norm $\| \cdot \|_p$ for some $p\in [1, + \infty]$. Let $A$ be a convex set in $\mathbb{R}^n$ and ...
- 489
0
votes
2
answers
230
views
Basic sequences in $ L_{p}$
Let $(x_{n})_{n}$ be a normalized basic sequence in $X=L_{p}$, with $1<p<2$.
Does there exist a subsequence $(x_{k_{n}})_{n}$ of $(x_{n})_{n}$ and a weakly null sequence $(x^{*}_{n})_{n}$ in $X^...
- 31.5k
2
votes
0
answers
106
views
Type-cotype inequalities for arbitrary orthonormal systems
Let $X$ be a B-convex Banach space and let $v^1 = (v^1_1,…,v^1_n), …, v^n = (v^n_1,…,v^n_n)$ be an orthonormal basis of $\mathbb{R}^n$. My question is what one can say about $\left( \sum_i \Vert \...
- 1,163
4
votes
0
answers
610
views
Does every separable Banach space have a Markushevich–Auerbach basis?
Let $X$ be a separable Banach space and $X^*$ be its dual, let $\{x_i\}$ be a sequence in $X$ with dense linear span and such that there exists a sequence $\{x_i^*\}$ in $X^*$ satisfying $x_i^*(x_j)=\...
- 4,713
11
votes
2
answers
6k
views
Is the $L^1$-space dual to a Banach space
Let $(\Omega,\mu)$ be a measure space. It is well known that for $1<p\leq \infty$ one has the duality
$$L^p=(L^{p*})^*,$$
where $1/p+1/p^*=1$.
Question. Is it known that the Banach space $L^1$ is ...
- 11.3k
2
votes
2
answers
645
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 ...
CommunityBot
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15
votes
1
answer
889
views
Operator norms of circulant matrices
The definition and basic properties of circulant matrices can be found here: http://en.wikipedia.org/wiki/Circulant_matrix.
For complex numbers $a_1,\ldots,a_n$, I will use the notation
$$
\mbox{...
- 1,167
5
votes
0
answers
138
views
Banach spaces complemented in their ultrapowers
By the principle of local reflexivity, the second dual $X^{**}$ of a Banach space $X$ is complemented in some ultrapower $X^U$ of $X$. Even when $X$ is separable, the index set of $U$ cannot be ...
- 11.3k
2
votes
1
answer
746
views
A unital algebra with norm and continuous multiplication is a Banach algebra
In my research in functional analysis, I came across this rather simple result:
For a normed algebra A over $ \mathbb{C} $ with unit, in which multiplication , right and left are both continuous w....
CommunityBot
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8
votes
2
answers
630
views
Extracting subsequences in Banach spaces, along an ultrafilter?
There are various principles in Banach space theory that allow one to pass from a given sequence of vectors $(x_n)$, to a subsequence $(x_{n_k})$ with some desired property. I'm thinking here, in ...
CommunityBot
- 1
5
votes
0
answers
186
views
Norm of projection onto functions of mean zero
Let $X$ be a finite set and consider the space $\ell^2(X;Y)$ of functions $\zeta:X\to Y$, where $Y$ is a fixed Banach space. It decomposes into a direct sum of constant function and its complement $\...
- 51
2
votes
1
answer
108
views
Sequences in $L_{p}(1<p<\infty)$ that is equivalent to the unit vector basis of $l_{p}$ or $l_{2}$
Let $1<p<\infty$. Johnson and Schechtman (Multiplication operators on $L(L_{p})$ and $l_{p}$-strictly singular operators, 2008, DOI: 10.4171/JEMS/141, eudml, arxiv) observed that if $(x_{n})_{n}$...
- 4,713
3
votes
0
answers
243
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A universal operator between separable Banach spaces
The Banach space $C[0,1]$ is universal for all separable Banach spaces in the sense that for a separable Banach space $X$ there is an isometric isomorphism from $X$ into $C[0,1]$. My question is ...
- 1,073
3
votes
1
answer
142
views
Subspaces of $L_{p}(2<p<\infty)$
Let $p>2$ and $X$ a subspace of $L_{p}$.
Then Kadec and Pelczynski proved that either $X$ is isomorphic to $l_{2}$ or $X$ contains a subspace isomorphic to $l_{p}$.
Question: if $X$ is ...
- 66.3k
5
votes
1
answer
400
views
Renorming a Banach space to make projections contractive
Let $X$ be a Banach space and $P$ be a projection in $B(X)$. Then $X$ can be renormed so that $P$ has norm $1$.
Can the same be done for a family of projections? That is, given finitely many ...
- 73.2k
21
votes
2
answers
1k
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 ...
CommunityBot
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10
votes
1
answer
439
views
Interpolation between $L_1^0$ and $L_2^0$
Let $L_p^0$ be the mean zero functions in $L_p(G)$, where, say, $G$ is an infinite compact group endowed with normalized Haar measure. Suppose that $T$ is a bounded linear operator on $L_1$ that maps $...
- 11.6k
3
votes
2
answers
337
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 ...
- 1,591
5
votes
1
answer
1k
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 ...
- 121
4
votes
0
answers
171
views
quasi-nilpotent part of a dual operator
Definitions and notation.
Let $X$ be a complex Banach space and $T\in\mathcal{L}(X)$ a continuous linear operator on $X$. We define the quasi-nilpotent part of $T$ as
\begin{equation*}H_0(T):=\left\{...
- 1,591
7
votes
1
answer
509
views
Davis, Figiel, Johnson and Pełczyński factorization through spaces with a bases
Davis, Figiel, Johnson and Pełczyński's Factorization Theorem states that each weakly compact operator $T:X \to Y$ between Banach spaces $X$ and $Y$ factors through a reflexive Banach space $Z$. In ...
- 31.5k
3
votes
1
answer
139
views
On the complemented subspaces of $L_{p}(p>2)$
M.I. Kadec and A. Pełczyński proved that if $E$ is a subspace of $L_{p}(p>2)$ isomorphic to $l_{2}$, then $E$ is complemented in $L_{p}$. My question is:
Is there a constant $C_{p}$ depending only ...
- 4,713
2
votes
0
answers
201
views
Reflexive subspaces of dual spaces
If $X$ is a Banach space, is it true that $X^{*}$ must contain an infinite dimensional reflexive subspace? I know of Gowers' example of a Banach space not containing $c_0$, $l_1$, or reflexive, but I ...
- 1,361
5
votes
2
answers
516
views
Biorthogonal functionals
If $X$ is a separable Banach space and $(x_n)$ is a basic sequence, then we can define biorthogonal functionals $(x^{*}_n)$ in $X^{*}$ such that $x^{*}_n(x_k)=\delta_{nk}$.
What about conversely? If ...
- 31.5k
4
votes
1
answer
389
views
Trivial intersection of kernels
This is a follow up to the question: Biorthogonal functionals. A positive answer to that question implies a negative answer to this one.
If $X$ is a separable Banach space, can we find a basic ...
CommunityBot
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3
votes
2
answers
435
views
A possible norm on a subspace of $C^\infty([0,1])$?
I have posted the following question (with minimal differences) on MSE some days ago, without receiving a satisfactory answer, so let me try here again.
Take the vector space of infinitely ...
CommunityBot
- 1
5
votes
0
answers
204
views
quasi-weakly compact operators, co-ideals of operator ideals, and Banach spaces $X$ with $X^{**}/X$ separable
Throughout, $X$ and $Y$ will denote Banach spaces with $T\in\mathcal{L}(X,Y)$ (the space of continuous linear operators between $X$ and $Y$). We define the operator $\overline{T}\in\mathcal{L}(X^{**}/...
- 1,591
1
vote
1
answer
290
views
Various limits of the Christoffel Darboux Kernel
In a different thread, we stumbled upon the following question:
Given a continuous finite measure $w(x)dx$ on some interval $(a,b)$, $-\infty \leq a <b \leq \infty$, and the set of respective ...
CommunityBot
- 1
2
votes
1
answer
143
views
Equivalent ways to study a semilinear parabolic equation as a perturbed abstract Cauchy problem
I am considering the following abstract Cauchy problem on Banach space $X$:
\begin{cases}
u'(t)=Au(t)+\big(f(t)+Bu(t)\big),&t\in[0,T],\\
u(0)=x_0,
\end{cases}
Suppose $A$ generates an analyitc ...
- 503
2
votes
0
answers
319
views
Banach-Mazur distance from finite-dimensional subspaces of $\ell_p$ to the Hilbert space
I am reading a paper http://www.math.tamu.edu/~johnson/TF3.4.pdf by Bill Johnson and Andrzej Szankowski and having trouble grasping why
$d_n(Z_m) \leq d_n(\ell_{p_{m+1}} ) = n^{|p_{m+1}-2|}$ in the ...
CommunityBot
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4
votes
0
answers
509
views
Good reference for noncommutative $L^p$ spaces
I'm looking for good references to learn about $L^p$ spaces associated with von Neumann algebras. I already know about Uffe Haagerup's paper "$L^p$-spaces associated with an arbitrary von Neumann ...
- 3,574
5
votes
1
answer
504
views
Compact non-nuclear operators
I am not sure if this question makes sense, or if it is trivial, but does there exists an infinite dimensional Banach space (necessarily without the approximation property) such that no compact, non-...
- 31.5k
6
votes
0
answers
252
views
Constructing Extreme Points in Reflexive Banach Spaces
A theorem of Lindenstrauss and Phelps states that if $X$ is a separable reflexive Banach space then the unit ball of $X$, $Ba(X)$, has uncountably many extreme points. The proof goes by contradiction ...
- 1,073
8
votes
2
answers
590
views
Attempted Banachification of a space
In a recent blog post, Terry Tao mentioned the question of how to tell if a Hausdorff topological vector space admits a finer topological structure which happens to be the topology of a Banach space (...
CommunityBot
- 1
4
votes
0
answers
209
views
On the weakly sequential completeness of the dual of the James space $J$
Let me first introduce some definitions. Let $1\leq p\leq \infty$.
A sequence $(x_{n})_{n}$ in a Banach space $X$ is said to be weakly $p$-convergent to $x\in X$ if the sequence $(x_{n}-x)_{n}$ is ...
- 4,713
8
votes
2
answers
369
views
$l^1$ versus $l^2$
Is there an elementary proof of this Banach space fact?
If the Banach space $V$ is linearly isomorphic to $l^1$, then it does not isometrically contain euclidean spaces of arbitrarily large finite ...
- 52.3k
11
votes
2
answers
1k
views
Do non-stable Banach spaces exist?
Let $K$ be $\mathbb{R}$ or $\mathbb{C}$. A Banach space $X$ over $K$ is stable if $X\cong X\times K$. I encountered the following question in some papers in the sixties:
Is every infinite ...
- 11.3k
1
vote
0
answers
80
views
Uniform limit of m-homogeneous polynomials over compact subsets of a Banach Space
I am trying to solve problem 1.2.A from Mujica's book "Complex Analysis in Banach Spaces".
We denote by $\mathcal{P}_a(^mE;F)$ the space of all $m-$homogeneous polynomials from $E$ into $F$, ...
CommunityBot
- 1
5
votes
1
answer
602
views
Invariant probability on a unit ball of a Banach space
Let $\Gamma$ be a discrete group acting on an infinite-dimensional Banach space $X$ by linear isometries.
Is there a probability measure (non-atomic, not supported on a finite dimensional subspace) ...
- 11.6k
5
votes
0
answers
175
views
A Banach space with the BD property and without the weak Gelfand-Phillips property
A subset A of X is called Grothendieck if every operator T from X to $c_0$ maps A to a relatively weakly compact set.
A Banach space has the weak Gelfand-Phillips property (wGP) if every ...
- 53.3k
0
votes
1
answer
138
views
Extracting a subsequence for which $\sup_j \left\vert \text{supp}(x_{n_j}) \right\vert < \infty$
Let $X$ be a Banach space with basis $(e_n)_{n=1}^\infty$, and suppose that $(x_i)_{i=1}^\infty$ is a normalized block basic sequence of $(e_n)_{n=1}^\infty$. In addition assume that $(x_i)_{i=1}^\...
- 2,429
1
vote
2
answers
181
views
Where can I find some articles and lecture notes in renorming theory in Banach spaces? [closed]
I am really into renorming theory in Banach spaces especially, renorming in non-reflexive Banach spaces such that they have nice property, for example they have fixed point property,locally uniformly ...
- 902
13
votes
0
answers
323
views
Kolmogorov width for cartesian products
For an operator $T:X\to Y$ between Banach spaces with unit balls $B_X$ and $B_Y$ the sequence of Kolmogorov widths is
$$
\delta_n(T)=\inf\lbrace \delta>0: T(B_X)\subseteq \delta B_Y +L \text{ for ...
CommunityBot
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4
votes
1
answer
376
views
Is the topological dual of a Banach space weakly* closed in its algebraic dual?
The question is completely contained in the title :)
I can only add, that it is not difficult to give a counterexample for normed spaces, and also Banach-Steinhaus theorem implies the sequential ...
- 217
1
vote
2
answers
873
views
$C[0,1]$ is Banach-space isomorphic to $c_0(C[0,1])$
$c_0(C[0,1])$ is the $c_0$-direct sum of countably many $C[0,1]$.How to prove
$C[0,1]$ is Banach-space isomorphic to $c_0(C[0,1])$.
Here,Banach-space isomorphism means a bounded invertible operator ...
- 185
10
votes
1
answer
515
views
Complemented subspaces in the dual of James' space $J$
James' space $J$ is subprojective; i.e., every infinite dimensional (closed) subspace of $J$ contains an infinite dimensional subspace which is complemented in $J$. This fact can be found in Corollary ...
- 11.3k