All Questions
1,222 questions
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 ...
- 3,343
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
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 ...
- 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 ...
- 1,361
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 ...
- 3,388
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
7
votes
3
answers
442
views
Weak compactness in the James space and its dual
It is known that there are characterizations of weak compactness in most of classical non-reflexive spaces (e.g. $L_{1}$-spaces and $C(K)$-spaces). I wonder whether there are characterizations of weak ...
- 3,343
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 ...
2
votes
0
answers
346
views
When is the sum of complemented subspaces complemented?
Let $X$ be a Banach space.
Question. Suppose $X_1,...,X_n$ are complemented subspaces of $X$. When is the sum $X_1+...+X_n$ complemented? Further, suppose we know some projections $P_1,...,P_n$ onto $...
- 935
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-...
- 1,361
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
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 ...
- 3,574
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 (...
- 148k
4
votes
0
answers
508
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 ...
- 41
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 ...
- 3,343
8
votes
2
answers
368
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 ...
- 42.8k
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 ...
- 7,583
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 $F$, ...
- 111
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) ...
- 53
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}^\...
- 125
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 ...
4
votes
1
answer
318
views
Non-equivalence of admitting different types of bases in Banach spaces
Whenever a certain type of (Schauder) basis is defined, it is natural to ask where that type lies in the scheme of other types of bases. This involves finding counter-examples of one type of basis ...
- 1,591
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 ...
- 823
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 ...
- 5,529
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
$$
\delta_n(T)=\inf\lbrace \delta>0: T(B_X)\subseteq \delta B_Y +L \text{ for ...
- 16.4k
31
votes
0
answers
2k
views
Do there exist infinite-dimensional Banach spaces in which every bounded linear operator attains its norm?
Let $X$ be a Banach space, $L(X)$ the space of all bounded linear operators on $X$. We say that $A ∈ L(X)$ attains its norm if there exists $x ∈ X$ such that $\|x\| = 1$ and $\|Ax\| = \|A\|$. The ...
- 4,878
1
vote
2
answers
873
views
$C[0,1]$ is Banach-space isomorphic to $c_0(C[0,1])$
$c_0(C[0,1])$ is the $c_0$-direct sum of countably many $C[0,1]$.How to prove
$C[0,1]$ is Banach-space isomorphic to $c_0(C[0,1])$.
Here,Banach-space isomorphism means a bounded invertible operator ...
- 185
15
votes
1
answer
441
views
Weak*-closure of finite rank operators on dual space
Given a Banach space $X$, we consider the space $B(X^*)$ of bounded, linear operators on $X^*$ with the weak*-topology from its canonical predual $B(X^*)_*=X^*\hat{\otimes}X$. What is $\overline{F(X^*)...
- 3,497
4
votes
0
answers
114
views
Weakenings of the Bounded Approximation Property
Let $X$ be a Banach space, and let $F(X)$ be the finite-rank operators on $X$.
Then $X$ has the $\lambda$-BAP if there exists a net $(S_i)_i$ in $F(X)$ with $\sup_i\|S_i\|\leq\lambda$ such that $...
- 3,497
29
votes
6
answers
9k
views
Nonseparable Hilbert spaces
Being nonseparable Banach space is in fact nothing special: one meets the first
examples in the standard functional analysis course, when one learns about
$\ell^p$ or $L^p[0,1]$ spaces-these spaces ...
- 9,330
2
votes
0
answers
552
views
Normed space between $H^{0+}$ and $L^2$
Cosider a function $f\in L^2(\mathbb{R}^3)$ with consider the following condition.
$$\int_{\mathbb{R}^3}\frac{|\widehat{f}(x)|}{1+|x|^{3/2}}dx<+\infty \, .\qquad\mbox{(*)} \, $$
Of course if $f\in ...
- 943
0
votes
1
answer
275
views
p-summable sequence
Let Y be a closed linear subspace of X and suppose that Y does not have copy of l1 .Does each weakly p-summable sequence in X/Y has a subsequence that's the image of a weakly p-summable sequence in X ...
- 29
12
votes
3
answers
564
views
Description of $\big(\ell^\infty(\mathbb N)\big)^{\!*}$ via ultrafilters
Let $\beta\mathbb N$ is the set of ultrafilters on $\mathbb N$ and $\mathscr F\in\beta\mathbb N$. Assume that $l_{\mathscr F}\in\big(\ell^\infty(\mathbb N)\big)^{\!*}$ is the functional which assigns ...
- 2,933
1
vote
0
answers
182
views
The real method of interpolation and operator ideals
Let $\overline{A} \mbox{ and } \overline{B}$ be n+1-tuples of Banach spaces and $T:\overline{A}\rightarrow \overline{B}$ be an interpolation operator; let $J(\overline{A})$ and (the corresponding, ...
0
votes
1
answer
179
views
Dense subspaces of $L^p(0,T;X)$
Given a Banach space $X$ and $1\leq p<\infty$, let's define the space $L^p(0,T;X)$ as the set of all strongly measurable functions $f:(0,T)\mapsto X$ such that
$$\int_0^T\Vert f\Vert_{X}^pdt<\...
- 1
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 ...
- 4,461
1
vote
1
answer
231
views
Hermitian Projections on $C[0,1]$
If $X$ is a normed linear space and $S(X)$ its unit sphere, $X′$ its dual space and $Π=\{(x,f)∈S(X)×S(X′) \ | \ f(x)=1\}$, then for an operator $T$ on $X$, the numerical range $V(T)$ is defined by $V(...
- 45
1
vote
2
answers
165
views
Antiproximanal subspace of $L_1[0,1]$
Could someone give a reference or construct an example of closed subspace of $Y\subset L_1[0,1]$ such that $\operatorname{dist}(x,Y)$ is not attained of for any $x\notin Y$.
I read somewhere that $Y$ ...
- 1,697
7
votes
1
answer
548
views
Spectrum of unitary elements of a Banach algebra
Unitary elements of a Banach space have been defined in this paper as follows:
Let $A$ be a Banach space and $a\in A, \|a\|=1$. Let $S_{a}=\{f\in A':\|f\|=1=f(a)\}$. Then $a$ is said to be (...
- 173
1
vote
0
answers
85
views
Coersivity of a bilinear form [closed]
I need to proof the coersivity of the following bilinear form. a,b and c are scalars, u is the velocity vector field and p is the pressure. Any help is much appreciated!
$$ B(\textbf{u},\textbf{v}) = ...
- 11
12
votes
1
answer
575
views
Is $\ell_p$ $(1<p<\infty)$ finitely isometrically distortable?
Let $Y$ be a Banach space isomorphic to $\ell_p$, $1<p<\infty$. Is it true that any finite subset of $\ell_p$ is isometric to some finite subset of $Y$?
It seems to me that it is an interesting ...
- 4,878
7
votes
1
answer
504
views
Is $L_q(X^*)$ complemented in $(L_p(X))^*$?
Let $X$ be a Banach space and let $p\in (1,\infty)$. If $q$ denotes the conjugate exponent to $p$, then $L_q(X^*)$ is easily seen to be isometric to a subspace of $(L_p(X))^*$ via the map $$f\mapsto \...
- 11.3k
7
votes
1
answer
469
views
Embedding of real trees into $\ell_1(\Gamma)$
It seems plausible that any real tree or ${\mathbb{R}}$-tree in the sense of the definition in https://en.wikipedia.org/wiki/Real_tree admits an isometric embedding into the Banach space $\ell_1(\...
- 4,878
3
votes
1
answer
807
views
Under what conditions can we put a complete norm on a linear subspace of a separable Banach space?
Question 1 Let $X$ a separable Banach Space and $Y\subset X$ linear subspace. When can we put a norm on $Y$ in such a way so that $Y$ is a Banach space?
Clearly if $Y$ is closed in the norm topology ...
- 243
3
votes
1
answer
280
views
example of an $\ell_1$-saturated Banach space without an unconditional basis
Giorgos Petsoulas, in his paper "A class of $\ell^p$ saturated Banach spaces," has constructed for each $1<p<\infty$ a space $\mathfrak{X}_p$ which is complementably $\ell_p$-saturated but ...
- 1,591
12
votes
3
answers
1k
views
Banach spaces $X$ with $\ell_2(X)$ not isomorphic to $L_2([0,1],X)$
Let $X$ be a Banach space. I think that some time ago I read somewhere that, in general, the space $\ell_2(X)$ of all sequences $(x_n)$ in $X$ with $\sum_{n=1}^\infty \|x_n\|^2<\infty$ is not ...
- 4,461
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
12
votes
2
answers
847
views
When is the closed unit ball in a smaller Banach space closed in a larger Banach space?
Recently I saw an interesting lemma:
For any $s>0$, the closed unit ball in $H^s$ is also closed in the $L^2$ norm. That is, suppose $u_j\in H^s$ and $\|u_j\|_{H^s}\le 1$. Suppose $u_j\to u$ in $L^...
- 5,169
0
votes
0
answers
123
views
On the operators from $l_{p}$ into Tsirelson's space $T$
Let $1<p<2$. My question is: Is any operator from $l_{p}$ into Tsirelson's space $T$ compact?
- 3,343