All Questions
369 questions with no upvoted or accepted answers
6
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132
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Mazur-Ulam bases in finite-dimensional Banach spaces
Definition. A basis $e_1,\dots,e_n$ of a finite-dimensional Banach space $X$ is called Mazur-Ulam if all vectors $e_1,\dots,e_n$ have norm one and every self-isometry $f:S_X\to S_X$ of the unit sphere ...
- 4,535
6
votes
0
answers
158
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Quotients of subspaces of $C(\alpha)$
A well known problem, attributed to H. P. Rosenthal, asks whether or not every quotient of $C(\alpha)$, $\alpha$ countable ordinal, is $c_0$-saturated. As it is known, $C(\alpha)$ are $c_0$-saturated ...
- 986
6
votes
0
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99
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Is every separable Banach space with the MAP 1-complemented in a space with a monotone basis?
The question, already phrased in the title, looks like a classical problem from Banach space theory from the 1970s. Hence, my question is more of a reference request in its nature.
Can every ...
- 11.3k
6
votes
0
answers
113
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Interpolation of some Sobolev spaces
Let $X_0=L^2(0,1)$, $X_1=H^4(0,1)$, $X_2=H^4(0,1)\cap H^2_0(0,1)$. We know the interpolation space $$(X_0,X_1)_{1/2,2}=H^2(0,1).$$
I am wondering what is
$$(X_0,X_2)_{1/2,2}=?$$
Would it be $H^2_0(0,...
- 395
6
votes
0
answers
252
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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
6
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0
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161
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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
6
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365
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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
6
votes
0
answers
307
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a question about Tsirelson's space
NOTE: I asked this question over at math.stackexchange.com but got no answer or comments after 3 days, probably because it's a bit specialized. Hopefully it is interesting enough to ask over here.
...
- 1,591
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
6
votes
0
answers
175
views
Ultrapowers and complemented subspaces
Let $Y$ be a closed subspace of a Banach space $X$, and let $\mathcal{U}$ be a nontrivial ultrafilter on the set $\mathbb{N}$ of all integer numbers.
It is not difficult to see that if $Y$ is ...
- 4,461
6
votes
0
answers
262
views
Given that a conditional measure is Gaussian, how bad can the original measure be?
Let $X$ and $Y$ be Banach spaces, and let $\varphi : X \to Y$ be a continuous linear map. Suppose that $\mathbb P$ is a probability measure on $X$ which satisfies the continuous disintegration ...
- 8,512
6
votes
0
answers
257
views
What is the intersection of the closures of left invertible operators and right invertible operators?
From Douglas Zare's answer (see Does $X$ embed in $Y$, and $Y$ embed in $X$, always imply that $X$ isomorphic onto $Y$?), one know that
$$ \overline{G_{l}(X,Y)} \bigcap \overline{G_{r}(X,Y) } = \...
- 515
6
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0
answers
484
views
Square and cube?
Gowers and Maurey proved in their remarkable paper(s), that there is a Banach space $X$ such that $X$ is isomorphic to its cube $X\oplus X\oplus X$ but not to isomorphic to its square $X\oplus X$. ...
- 113
6
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answers
3k
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Projective and injective tensor product
It is well known that for arbitrary Banach spaces $X$ and $Y$ we have that the dual space
$(X \hat{\otimes}_{\pi} Y)^* = \mathcal{L}(X, Y^*)$.
If we take $\ell^p$ and $\ell^q$ such that $p < q^{\...
6
votes
0
answers
2k
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Weak lower semi-continuity
Which conditions assure the weak lower semicontinuity of, say, an integral functional of the type
$F(u):=\int_\Omega f(u(x),Du(x))dx$ on $W^{1,2}(\Omega,\mathbb{R}^N)$ for a bounded, if you will even ...
6
votes
0
answers
639
views
Hilbert subspaces of indefinite inner product spaces
Let $E$ be a real linear space, endowed with a non-degenerate symmetric
bilinear form $(.,.)$.
Suppose that the [indefinite] inner product space $(E,(.,.))$
satisfies the following [sequential] ...
- 4,060
5
votes
0
answers
899
views
Is such a Banach space $X$ isometrically isomorphic to a Hilbert space or not?
Let $X$ be a real or complex Banach space. $X$ satisfies:There exists real or complex series $\{a_k\}_{k=1}^n,\{b_k\}_{k=1}^n$ (which satisfies that:$\begin{cases}
a_k,b_k\in \mathbb{R}\ \forall k=1,\...
- 181
5
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137
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A list of properties of $(\bigoplus \ell^1_n)_{\ell^p}$, $1<p<\infty$
The Banach space $E=(\bigoplus_{n=1}^{\infty}\ell^1_n)_{\ell^p}$ for $1<p<\infty$ shows up in various places in the literature to construct counterexamples. The purpose of this post is to ...
- 2,605
5
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answers
265
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Automorphic Banach spaces
A Banach space $X$ is called automorphic if for every closed subspace $Y\subseteq X$ with $\dim X/Y=\infty$, every automorphism (= linear continuous isomorphism) of $Y$ can be extended to an ...
- 4,535
5
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204
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Given a totally ordered system of Banach spaces, can we we always change the norms to get isometric embeddings?
Given a real vector space $V$ which is the union of a totally ordered family of vector subspaces $V=\bigcup_{i\in I} V_i$. By that I mean that we assume that $(I,\leq)$ is a totally ordered set and ...
- 609
5
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answers
169
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Is the Grassmannian of a Banach space an infinite dimensional manifold?
Grassmannian of complemented subspaces in a Banach space is a Banach manifold. This is explained for example in the thesis of Douady and is rather analogous to the finite-dimensional case.
I would ...
- 344
5
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0
answers
280
views
Completeness of the space $L^p$ and the Axiom of Countable Choice
I am thinking about the proof that the usual $L^p$ spaces are complete.
So, let $(X,\mathcal{F},\mu)$ be a measure space and let
$p\in[1,+\infty)$.
Important: by a measure I mean a nonnegative $\sigma$...
- 935
5
votes
0
answers
315
views
Schauder basis in the Arens-Eells space
Context
Arens-Eells space. Let $X$ be a separable pointed metric space with base point $e$. An elementary molecule is defined as follows (Nik Weaver, Lipschitz Algebras, 2nd ed.)
$$
m_{pq} := \delta_p ...
- 437
5
votes
0
answers
145
views
Second dual $X^{**}$ of ternary $C^*$-ring $X$ is again ternary $C^*$-ring?
Recall that a ternary $C^*$-ring is a complex Banach space $X$, equipped with a associative ternary product $[.,.,.]:X^3 \to X$ which is linear in outer variables and conjugate linear in middle ...
- 1,115
5
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0
answers
199
views
Standard function spaces with the approximation property
A Banach space $\mathcal{X}$ is said to have the approximation property (AP) if, for every compact set $K \subset \mathcal{X}$, there is a sequence of finite rank operators $\{U_n : \mathcal{X} \to \...
- 313
5
votes
0
answers
139
views
Copies of $\ell_\infty^k$ in subspaces of the space of operators between $n$-dimensional Banach spaces
Are there a positive integer $k$ and an unbounded increasing function $d:\mathbb N\to\mathbb N$ (of growth order $\Omega(n^2)$) such that for any $n$-dimensional Banach spaces $X,Y$, the Banach space $...
- 4,535
5
votes
0
answers
350
views
How to calculate the volume of a parallelepiped in a normed space?
Let $E$ be a real normed space, and let $v_1,...,v_n\in E$ be linearly independent. The parallelepiped defined by these vectors is $P=\{\sum_{i=1}^{n}\alpha_i v_i|~0\le\alpha_i\le 1\}$. Since $E$ is a ...
- 5,529
5
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228
views
What is the smallest number of hyperplanes covering $\ell_2$?
For a Banach space $X\ne \{0\}$, let $\mathrm{cov}_H(X)$ be the smallest number of hyperplanes covering $X$.
By a hyperplane in a Banach space I understand any closed affine subspace of codimension ...
- 41.9k
5
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231
views
Which subspaces of $\ell_p^n$ are isometric?
This question is similar to the one asked here:
Extending linear isometries from subspaces of $\ell_p^n$
Let $p$ be an even integer. Let $X,Y$ be subspaces of $\ell_p^n$, and let $U : X \to Y$ be a ...
- 53
5
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0
answers
134
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Banach space properties defined by compact operators, strictly singular operators and strictly cosingular operators
Let $X,Y$ be Banach spaces. We denote by $\mathcal{L}(X,Y)$ the space of all operators from $X$ into $Y$, $\mathcal{K}(X,Y)$ by the space of all the compact operators from $X$ into $Y$, $S(X,Y)$ by ...
- 3,343
5
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0
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245
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Examples of Banach lattices with positive Schur property but without Schur property
A Banach lattice $E$ has the
$(1)$ Schur property provided that any weakly null sequence in $E$ is norm null in $E$, and
$(2)$ positive Schur property provided that any weakly null sequence of ...
user114263
5
votes
0
answers
345
views
Weak to weak$^*$ continuity of the duality mapping
Let $X$ be a uniformly convex and uniformly smooth Banach space. We consider the duality mapping $J_p^X$ defined as the sub-gradient $\partial (\frac1p\|\cdot\|^p)$. Is there a characterisation of the ...
- 799
5
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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
5
votes
0
answers
150
views
On the relation between Lipschitz free-spaces
Let $X$ be a pointed metric space, with base point 0. The space of Lipschitz function which preserves the base point,
$Lip_0(X)=\{f:X\to\mathbb{R} : f(0)=0\}$ consider with the norm $\|f(x)\|=\sup_{x\...
- 71
5
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0
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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
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
5
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0
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204
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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
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 ...
5
votes
0
answers
206
views
On a variant of Eidelheit's theorem
A theorem of Eidelheit from 1940's asserts that two Banach spaces $X$ and $Y$ are isomorphic if and only if $L(X)$ and $L(Y)$, the algebras of all bounded linear operators, are isomorphic as Banach ...
- 51
5
votes
0
answers
2k
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Denseness of finite rank operators in $\mathcal{B}(X,Y)$
Let $X$ and $Y$ be Banach spaces and let $\mathcal{B}(X,Y)$ be the space of bounded linear operators from $X$ to $Y$. As noted in the answers to a question on
https://math.stackexchange.com/questions/...
- 107
5
votes
0
answers
133
views
Banach spaces admitting no proper quasi-affinity
I am interested in examples of Banach spaces $X$ satisfying the following two conditions:
(1) Every (continuous linear) injective operator $T:X\to X$ with dense range is surjective.
(2) $X$ is ...
- 4,461
5
votes
0
answers
598
views
Do the banded operators check the invariant subspace problem?
Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators.
Invariant subspace problem: Let $T \in B(H)$. Is there a non-trivial closed $T$-invariant ...
5
votes
0
answers
200
views
almost projective Banach space, complex scalars
It is well-known that if a real Banach space $E$ is "almost metrically projective" then $E$ is isometrically isomorphic to some $\ell^1(\Gamma)$. We say $E$ is "almost metrically projective" if ...
- 1,704
5
votes
0
answers
425
views
Reflexive-saturated Banach spaces
Say that a Banach space $X$ is strongly saturated by reflexive subspaces if every closed subspace $Y\subset X$ contains a further reflexive subspace $Z\subset Y$ with $\mbox{dens }Y=\mbox{dens Z}$. If ...
- 387
4
votes
0
answers
148
views
Some questions on Hardy's spaces
In the paper http://www.numdam.org/item/CM_1976__33_3_261_0.pdf, the authors have asked in Page 285 whether the Hardy space $H^p$ embeds isometrically into the Hardy space $H^q$ for $1\leq q<p<...
- 1,879
4
votes
0
answers
108
views
Larger possible chain of closed subspaces in the dual of a Banach space
In this question, is demonstrated that a separable space can have a chain (ordered by inclusion) of closed subspaces with uncountable many subspaces.
My question is the following. If $X$ is an ...
- 153
4
votes
0
answers
73
views
Find reasonable definition for endpoint Lorentz function spaces $L^{\infty,q}$ via the idea from endpoint Triebel-Lizorkin ${\scr F}_{\infty,q}^s$
On a measure space $(X,\mu)$, for $0<p,q<\infty$ the Lorentz space $L^{p,q}(\mu)$ is defined by $$\|f\|_{L^{p,q}(\mu)}:=p^\frac1q\|t\mu(|f|>t)^\frac1p\|_{L^q(\mathbb R_+,\frac{dt}t)}=p^\...
- 938
4
votes
0
answers
116
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Weakly null sequences in projective tensor products II
The question in this post is the question below from an article by Rodriguez & Rueda Zoca [1].
Below is a complimentary salad/side dish that accompanies the main course.
Let $B^2(X,Y)$ denote ...
- 2,605
4
votes
0
answers
149
views
Isomorphic copies of $c_0$ in the projective tensor products
There exist Banach spaces $X$ such that the projective tensor product $X\mathbin{\hat{\otimes}}_\pi X$ contains an isomorphic copy of $c_0$ [BourgainPisier1983]. Moreover, $X$ is an $\mathcal{L}_\...
- 2,605
4
votes
0
answers
260
views
On the predual of the James tree space $\mathit{JT}$
$\newcommand\JT{\mathit{JT}}$The James tree space $\JT$ was the first example of a separable Banach space containing no copies of $\ell_1$ such that its dual space is non-separable. Since $\JT$ admits ...
- 4,461