All Questions
1,222 questions
8
votes
1
answer
434
views
Self-dual finite-dimensional complex normed spaces
Suppose $X$ is a complex normed space of dimension 2 or 3 and $X$ is isometrically isomorphic to its dual. Is $X$ a Hilbert space?
Remarks: There are easy counterexamples in the real case, and in ...
- 11.4k
5
votes
1
answer
345
views
Location of a Banach Space inside its bidual
Let $X$ be a Banach Space and let $Y$ be a closed subspace of $X^{**}$ such that $X\bigcap Y=0$. Let $P$ be the quotient map from $X^{**}$ onto $X^{**}/ Y$. I need to prove or refute that $P\left|_{X}\...
- 5,529
9
votes
0
answers
261
views
SVD-type decomposition for the tensor product of three Hilbert spaces?
(The questions How does the Schmidt decomposition generalize to tensor products of several finite-dimensional systems? and Is there a useful generalization of the Schmidt decomposition to the ...
- 25.8k
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
4
votes
0
answers
144
views
Embedding of $\ell_2$ in $L^p([0,1])$
Let $(g_n)_{n\geq 1}$ be a sequence of i.i.d. complex Gaussian random variables on $[0,1].$ Then it is easy to see that the map $j:\ell_2\to L^p([0,1])$ defined as $je_n=[E(g_n^p)]^{\frac{1}{p}}g_n,n\...
- 455
2
votes
0
answers
124
views
Logarithm of $L^p$ space
I encountered the following space as a natural space for setting up a certain problem:
$$
S_m^p = \{f \colon I \to \mathbb{R} \text{ measurable }; m^{f} \in L^p(I)\}
$$
Here, $I$ is an open bounded ...
- 648
6
votes
1
answer
253
views
Is every ideal part of an operator ideal?
An operator ideal $\mathfrak J$ is a class of continuous operators. Namely, for every pair of complex Banach spaces, $\mathfrak X,\mathfrak Y$, we have that $\mathfrak J(\mathfrak X,\mathfrak Y) \...
- 3,984
6
votes
1
answer
223
views
Is the space of vectorial functions that are Dunford integrable complete?
Let $X$ a Banach Space and $(\Omega, \Sigma, \mu)$ a measure space. A function $F:\Omega\rightarrow X$ is Dunford integrable if $x^\ast\circ F$ is $\mu$-integrable for every $x^\ast\in X^\ast$. The ...
3
votes
1
answer
151
views
The weakest condition guarantees some Separation-type of convex sets in Banach spaces
Classical Hahn-Banach Separation theorem plays a vital role in many branches of Analysis, Like functional Analysis, Convex Analysis, Variational Analyis, Theory of ODEs, optimal control and ...
- 369
0
votes
0
answers
263
views
Does AX+XA=0 have any non-trivial solutions?
Let $X$ be a continuous linear self-adjoint operator on some Hilbert space $H$ and for arbitrary compact operators $A$ we have: $XA+AX=0.$ Does this imply that $X=0$ or can there be non-trivial ...
- 305
2
votes
0
answers
93
views
Open problems concerning Araujo's biseparating maps
Araujo stated the following four open questions at the end of his paper, page $518$ and $519.$
Question $1:$ Assume that there exists a biseparating map $T:A^n(\Omega:E)\to A^m(\Omega',F)$ which is ...
- 623
4
votes
1
answer
283
views
Absolutely continuity in variation of constant formula
We are talking here about the initial value problem on some Hilbert space $H$
$$y'(t)=Ay(t)+f(t), \\ y(0)=y_0 \in D(A).$$(Problem 1.13 in the reference)
Then $y(t)=e^{At}y_0 + \int_0^t e^{A(t-s)}f(s) ...
- 43
7
votes
1
answer
433
views
Extending compact operators
Let $X$ be a separable, infinite-dimensional complex Banach space and $Y\subseteq X$ an infinite-dimensional closed subspace. Suppose $K:Y\to X$ is an arbitrary compact operator. I would like to ...
- 1,591
10
votes
0
answers
207
views
Projective tensor squares of uniform algebras
In discussion with a colleague recently (Jan 2017),
$\newcommand{\AD}{A({\bf D})}\newcommand{\CT}{C({\bf T})}$
I was reminded that if $A(D)$ denotes the disc algebra and $\iota: \AD\to \CT$ is the ...
- 25.8k
1
vote
1
answer
373
views
does every compact convex set in c0 have but countably many extreme points
This seems plausible, given the properties of the unit ball of $c_0$.
I have a compact set in a complex Banach space $X$ whose closed convex hull has uncountably many extreme points. It would be ...
- 1,591
5
votes
1
answer
587
views
Infinite dimensional subspaces of $L^1$
Suppose that $X$ is an infinite dimensional subspace of $L^{1}$. In some cases it is true that $X$ contains an isomorphic copy of an infinite dimensional Hilbert space. However, it is not the case ...
- 5,005
5
votes
1
answer
109
views
Simultaneous near-best approximation with respect to two norms
Suppose that $M$ is a closed infinite dimensional subspace of $L_4(0,1)$ which is also a closed subspace of $L_1(0,1)$. Hence $M$ is isomorphic to $\ell_2$ as a subspace of $L_p(0,1)$ for $1\leq p\leq ...
- 4,461
1
vote
1
answer
52
views
Infinitely many independent functions that are only frequency localized?
A function $f \in L^2(\mathbb R^d)$ will be called $K$-frequency localized if the following inequality holds
$$\int_{\mathbb R^d} \lvert \widehat{f}(x) \rvert^2 x^2 \ dx \le K \int_{\mathbb R^d} \...
- 11
3
votes
1
answer
154
views
Uncomplemented copies of $\ell_2$ in $L_p(0,1)$ for $2-\varepsilon<p<2$
It was proved in [Bennett, G.; Dor, L.E.; Goodman, V.; Johnson, W.B.; Newman, C.M. On uncomplemented subspaces of $L_p$, $1<p<2$. Israel J. Math. 26 (1977), 178–187]
that, for $1 <p < 2$, ...
- 4,461
4
votes
2
answers
311
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 ...
- 5,005
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
3
votes
0
answers
89
views
Discrete Lions Peetre interpolation
In S. Heinrich's "Closed operator ideals and interpolation," Heinrich describes two equivalent descriptions of Lions-Peetre interpolation space $(X,Y)_{\theta, p}$ for $0<\theta<1$ and $1\...
user114263
4
votes
2
answers
558
views
Is a specific sequentially closed subset of $M([0,1])$ closed?
Let $M([0,1])$ be the set of finite signed measures on $[0,1]$
(with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all $\...
- 95
1
vote
1
answer
221
views
Kernel of a convex combination of projections
Assume that we have finitely many projections $P_1,\dots, P_n$ on a Banach space $X$ (take an explicit case of $X=L_p(\mu)$ if a concrete examples is better). Consider their convex combination $P=\...
- 11
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
5
votes
0
answers
103
views
Complementation problem for $\ell_p^2$
Let $n\in\mathbb{N}$ and $p,q\in(1,+\infty)$ with $p^{-1}+q^{-1}=1$. Consider isometric embedding between $\mathbb{C}$-Banach spaces
$$
\rho:\ell_p^n\to\ell_\infty(S, \ell_1^n),x\mapsto(f\cdot x)_{f\...
- 1,697
4
votes
1
answer
215
views
almost invariant half space for a dual of a restricted operator
Let $X$ be an infinite-dimensional Banach space and $T\in\mathcal{L}(X)$ a continuous linear operator acting on $X$. A closed subspace $Y$ of $X$ is said to be an almost-invariant halfspace (...
- 1,591
0
votes
0
answers
70
views
A question about an irreducible ultra-power
Let $A$ be a Banach algebra and $E$ be an irreducible Banach $A$-module. Is there a countably incomplete ultra filter $\mathcal U$ on $\mathbb N$, the set of natural numbers, such that the ultra power ...
- 2,118
9
votes
4
answers
1k
views
Boundedness of nonlinear continuous functionals
Let $K$ be the closed unit ball of $C[0,1]$, and let $f$ in $C(K,\mathbb{\, R})$.
Is it true that there exists an infinite dimensional reflexive subspace
$E$ of $C[0,1]$ s.t. $f(K\cap E)$ is bounded ?
...
- 4,060
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 ...
- 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
0
votes
0
answers
65
views
Does $\{ x^* \circ \psi_t:x^*\in ext(E^*), t\in K \}\subset ext(X^*)$ hold?
Notations: Let $K$ be a locally compact Hausdorff space and $E$ be a real normed linear space.
Recall that $C_0(K,E)$ is the set of $E$-valued continuous functions $f$ on $K$ such that $f$ vanishes at ...
- 623
4
votes
1
answer
391
views
Frechet differentiable implies reflexive?
Let $X$ be a Banach space. Is it true that if dual norm of $X^*$ is Frechet differentiable then $X$ is reflexive?
Can any one help me?
thanks
- 823
10
votes
1
answer
253
views
Approximation via finite rank Cameron-Martin projections
Let $(W, \|\cdot\|_W)$ be a real separable Banach space equipped with
a non-degenerate Gaussian Borel measure $\mu$. Let $H \subset W$ be
the corresponding Cameron-Martin Hilbert space (also known as ...
- 29.8k
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
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
3
votes
0
answers
169
views
A spanning set for an annihilator set on a Banach space
Let $(z_n)$ be a $H^\infty$-interpolating sequence on the open complex unit disc $\mathbb D$. If $A$ is some Banach space of analytic functions on the disc, denote by $X$ the closed subspace of all ...
- 31
5
votes
2
answers
353
views
Contractively complemented subspaces of $c_0(I)$
Does every contractively complemented subspace of $c_0(I)$ is isometric to $c_0(J)$ for some $J\subseteq I$?
May be someone has a counterexample?
- 1,697
1
vote
0
answers
996
views
The dual of the space of smooth functions that vanish at infinity
Let $U \subset \Bbb R ^n$ be an open subset and let $\mathcal C$ be the space of the smooth functions on $U$ that vanish at infinity, endowed with the seminorms $p_\alpha (f) = \sup \limits _{x \in U} ...
- 5,407
1
vote
0
answers
220
views
About the projection on the unit sphere
Let $H$ be a Hilbert Space and let $A\subset H$ be a connected set such that any two elements of $A$ are linearly independent and also $A^{\bot}=\left\{0\right\}$ (this seems to be immaterial). Is ...
- 5,529
8
votes
1
answer
455
views
Closure of $L(\ell^2,\ell^2)$ in $L(\ell^2,\ell^\infty)$
Let $\ell^2$, $\ell^\infty$ denote the usual sequence spaces and let $L(\ell^2,\ell^2)$ the Banach space of bounded linear operators from $\ell^2$ to $\ell^2$ as well as $L(\ell^2,\ell^\infty)$ the ...
- 261
10
votes
1
answer
929
views
Non-probabilistic proof of the Johnson–Lindenstrauss lemma
The Johnson–Lindenstrauss lemma states that a small set of points in a high-dimensional space can be embedded into a space of much lower dimension in such a way that distances between the points are ...
- 225
3
votes
1
answer
640
views
Relationship between LlogL and Hardy spaces
I think that for positive, one-dimensional, periodic functions, the following statement is true:
$$
f\in L log L(\mathbb{T})\Leftrightarrow f\in H^1(\mathbb{T}),
$$
where
$$
LlogL=\{f\in L^1\,s.t.\,\...
- 843
7
votes
3
answers
814
views
Preduals of B(E)
For a Hilbert space $H$ it is well known that the algebra $B(H)$ has a unique predual; the Banach space of trace class operators.
If $E$ is a Banach space then is it known whether
$B(E)$ is always a ...
- 1,411
3
votes
1
answer
153
views
Example of a strictly cosingular operator whose dual is not strictly singular?
The short version of my question: Suppose $T\in\mathcal{L}(X,Y)$ is strictly cosingular. Must $T^*$ be strictly singular?
The long version.
Let $X$ and $Y$ be Banach spaces, and denote by $\mathcal{...
- 1,591
11
votes
3
answers
1k
views
Continuous automorphism groups of normed vector spaces?
Consider the metric space on, say, ℝ2 induced by the various $L^p$ norms, and the group of isometries from that space into itself that preserve the origin. When $p=2$ I get the continuous group ...
- 180
3
votes
1
answer
657
views
Banach space of discontinuous functions(Killing continuous functions)
Edit: According to the comment of Prof. Majer, I revise the question:
For a metric space $X$, we put $A=\{f:X\to \mathbb{C}\mid \text{f is bounded}\}$. We define two semi norm on $A$
$$\...
- 356
10
votes
1
answer
591
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 ...
- 11.3k
3
votes
0
answers
125
views
Commutative discrete cyclic operator groups on topological vector spaces
Let $V$ be a complex Hausdorff separable topological vector space of infinite dimensions. Does there exist a commutative discrete subgroup $A\subset\mathcal{L}(V)$ of continuous operators on $V$ with ...
- 1,959
6
votes
1
answer
418
views
Reflexive subspaces of non-separable abstract $L_1$ spaces
An abstract $L_1$ space is a Banach lattice $E$ such that $\|x+y\|=\|x\|+\|y\|$ for disjoint $x,y\in E$. The space $L_1[0,1]$ is a separable example that contains subspaces isomorphic to $L_p[0,1]$ ...
- 4,461