All Questions
1,222 questions
11
votes
2
answers
513
views
What is the structure of a Banach space $X$ when $Y$ and $X/Y$ are hereditarily indecomposable?
Assume that $X$ is a separable Banach space and $Y$ a closed subspace such that
$Y$ and $X/Y$ are hereditarily indecomposable (HI). The general question is what is the possible structure of $X$.
...
- 986
11
votes
1
answer
487
views
Is the spectrum of a "self adjoint" operator real on $\ell^p$?
There might be an obvious answer to the question, but it doesn't come to mind.
Suppose we have an infinite matrix $A=(a_{ij})$, which defines a bounded linear operator on $\ell^p$, i.e. for all ...
11
votes
1
answer
336
views
Notions in the literature capturing the "symmetric" or "homogeneous" flavour of $L_p$?
This post/question is admittedly vague, but I hope that with some feedback in comments it could be made more precise.
For $E$ a Banach space, $K(E)$ and $B(E)$ will denote the Banach algebras of ...
- 25.8k
11
votes
1
answer
227
views
Complemented subspaces of $C(\beta\mathbb N\times \beta\mathbb N)$
Problem. Is there any complemented subspace in the Banach space $C(\beta\mathbb N\times\beta\mathbb N)$, not isomorphic to $c_0$, $c_0\oplus C(\beta\mathbb N)$, $C(\beta\mathbb N)$, $c_0(C(\beta\...
- 4,535
11
votes
0
answers
342
views
The diagonal operators and unconditionality
The following is well-known:
Theorem: Let $X$ be a Banach space with an unconditional basis $(e_n)_n$.
Then the space of the diagonal operators with respect the basis $(e_n)_n$ endowed with
the ...
- 986
11
votes
0
answers
266
views
Quantifier swap in Banach space theory
The uniform boundedness principle and its corollaries from a logical point of view are statements of when one can swap quantifiers in Banach spaces. Take for instance the principle of condensation of ...
- 301
11
votes
0
answers
389
views
Von Neumann Inequality in Banach spaces
It is known that the only Banach space that satisfies the von-Neumann inequality is the Hilbert space:
Theorem (see e.g. Pisier, "Similarity Problems and Completely Bounded Maps", p 27) For a Banach ...
- 5,529
11
votes
0
answers
626
views
Outline of Generic Separable Banach Spaces don't have a Schauder Basis
So, I know P. Enflo showed that there is a separable Banach Space that doesn't satisfy the approximation property. My professor mentioned during class that in fact generic separable Banach Spaces don'...
- 243
11
votes
0
answers
622
views
Subspaces and quotients in Banach space theory
In Banach space theory (closed) subspaces and quotient seem to play a symmetric role. However, since the behavior of subspaces is more intuitive, subspaces appear more frequently. E.g., the theory of ...
- 4,461
10
votes
2
answers
489
views
Surjective linear isometries on $\ell_\infty(\mathbb{N})$
In Volume 1 of "Classical Banach Spaces" Lindenstrauss and Tzafriri note that all surjective linear isometries on $\ell_\infty$ are of the from $(a_i) \mapsto (\varepsilon_i a_{\pi(i)})$ ...
- 1,073
10
votes
3
answers
739
views
Is there a version of Fischer-Riesz theorem for Banach space?
$( \Omega,F, P )$: a measurable space equipped with a finite measure
$(B , \Vert \cdot \Vert) $ : a Banach space with $\mathcal{B}$ as its borelian $\sigma$-algebra
$p$ : a constant bigger than $1$
...
- 406
10
votes
1
answer
509
views
A quantity measuring the separability of Banach spaces
Let $X$ be a Banach space. It is natural for us to introduce a quantity measuring the separability of sets as follows: for a subset $A$ of $X$, we set
$\textrm{sep}(A)=\inf\{\epsilon>0: A\subseteq ...
- 3,343
10
votes
2
answers
5k
views
Direct proof of the separation theorem of Hahn-Banach
The "extension" (or "analytic") form of the theorem of Hahn-Banach has a natural and yet elegant proof. In just any textbook I have ever seen, it is proved first; the "separation" (or "geometric") ...
- 3,388
10
votes
2
answers
881
views
volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$?
We denote by $\otimes_{\epsilon}$ the injective Banach tensor product.
Which is the asymptotic volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$?
- 1,222
10
votes
2
answers
666
views
Reference request: Extensions of Wiener's Tauberian Theorem
Wiener's Tauberian Theorem says that linear combinations of translations of a function $f$ are dense in $L^1(\mathbb{R})$ if and only if the zero set of the Fourier transform of $f$ is empty. This is ...
- 710
10
votes
1
answer
368
views
Group of isometries of Banach spaces a topological group?
Let $X$ be a Banach space and let $\mathrm{Iso}(X)$ be its group of isometries, i.e., the set of surjective linear maps $T: X \to X$ with $\|Tx\| = \|x\|$.
Q: Is $\mathrm{Iso}(X)$ a topological group ...
- 9,853
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
10
votes
2
answers
843
views
Implicit function theorem with continuous dependence on parameter
Let $X,Y$ be Hilbert spaces and $P$ a topological space$^1$ and $p_0\in P$.
Let $f:X\times P\to Y$ be a continuous map such that
for any parameter $p\in P$, $f_p:= f|_{X\times \{p\}}:X\to Y$ is ...
- 2,533
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 $...
- 31.5k
10
votes
1
answer
203
views
Verbal description, or terminology, for the ${\mathcal L}_p$-spaces of Lindenstrauss and Pelczynski
This question is intended for Banach-space specialists and so I will not repeat all the definitions here. My aim is to find out how the Banach space community refers to such spaces in discussions, and ...
- 25.8k
10
votes
1
answer
366
views
Are all compact subsets of Banach spaces small in a measure-theoretic sense?
Definition. A subset $K$ of a topological group $X$ is called measure-continuous if there exists a $\sigma$-additive Borel probability measure $\mu$ on $X$ such that for every compact subset $C\subset ...
- 41.9k
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
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
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
10
votes
1
answer
900
views
Approximation of a compactly supported function by Gaussians
Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function whose support is a closed interval, e.g. $\text{supp}(f)=[a,b]$. Then $f$ can be approximated (e.g. in $L^2$) by a linear combination of Gaussian ...
- 710
10
votes
2
answers
2k
views
Pull-back of generalized functions
Let $f\colon X\to Y$ be a smooth map between smooth manifolds. Then the pull-back operation
$f^*\colon C^\infty(Y)\to C^\infty(X)$ is a linear continuous operator when $C^\infty$ is equipped with the ...
- 21.8k
10
votes
0
answers
226
views
Extremal bases in finite-dimensional Banach spaces
Definition. A basis $e_1,\dots,e_n$ for a Banach space $X$ is called extremal if there exists a point $s$ in the unit sphere $S_X=\{x\in X:\|x\|=1\}$ such that for every $i\in\{1,\dots,n\}$ the ...
- 4,535
10
votes
0
answers
230
views
Norm-attaining operators with values in a 2-dimensional Hilbert space
Is the set $N\!A(X,\ell_2^2)$ of norm-attaining operators from a Banach space $X$ onto the $2$-dimensional Hilbert space $\ell^2_2$ dense in the Banach space $L(X,\ell_2^2)$ of all linear continuous ...
- 4,535
10
votes
0
answers
266
views
Are biduals of spaces of differentiable functions on hypercubes Grothendieck?
Consider the space $E_n = C^1([0,1]^n)$ of continuously differentiable functions with the usual norm
$$\max\{ \|f\|_\infty, \|f^\prime_{x_1}\|_\infty, \ldots, \|f^\prime_{x_n}\|_\infty\}.$$ making it ...
- 11.3k
10
votes
0
answers
251
views
Do sufficiently large Banach spaces admit non-compact operators with not too large range?
As in the title,
does there exist a cardinal number $\lambda$ such that for every Banach space $X$ of density/cardinality at least $\lambda$ there exists a non-compact bounded, linear operator $T\...
- 11.3k
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
10
votes
0
answers
609
views
Asymptotic non-distortion of the separable Hilbert space
By the work of E. Odell and Th. Schlumprecht, we know that the
separable Hilbert space $\ell_2$ is arbitrarily distortable. But
I don't know if an "asymptotic" version of their result is true.
To ...
- 576
9
votes
4
answers
4k
views
Is the space of Radon measures a Polish space or at least separable?
Background: I work on a SPDE problem where in order to apply Prokhorov's theorem I need that some measure space is Polish space. And additionaly it would be good if that space is Banach space. Earlier ...
- 657
9
votes
3
answers
4k
views
Projections in Banach spaces
Dear All,
I am absolutely lost in the following problem:
Let $P_s, \: s \in [0,1],$ be a uniformly bounded family of projections (idempotents) in a Banach space $X$ such that $P_s P_t = P_{{\rm min}...
- 93
9
votes
2
answers
338
views
Does $End(V)$ remember $V$, where $V$ is a locally convex space?
Let $V$ be a locally convex topological vector space over $\mathbb C$, and let $A=\mathrm{End}(V)$ be its algebra of continuous linear endomorphisms (viewed just as a $\mathbb{C}$-algebra, not as a ...
- 43.2k
9
votes
1
answer
1k
views
Counterintuitive consequences of the Hahn-Banach theorem
The axiom of choice has many counterintuitive consequences like the Banach-Tarski paradox.
The Hahn-Banach theorem is a consequence of the axiom of choice, but it is weaker.
I would like to know ...
- 4,461
9
votes
3
answers
868
views
Rosenthal like inequality for weak $\mathbb L^p$-norms
Let $p$ be a real number greater than $1$. It is well known (see Hall and Heyde's Martingale limit theory and its applications, Theorem 2.10) that there exists a constant $C_p$ such that if $(X_i)_{i=...
- 3,958
9
votes
1
answer
852
views
Banach space with uncountable basis
We know that an infinite dimensional Banach space has an uncountable Hamel basis. Now if $X$ is a vector space with an uncountable Hamel basis, does there exist a norm on $X$ for which $X$ is a Banach ...
- 585
9
votes
1
answer
333
views
Closedness of linear image of positive L1 functions
Let $\mathcal X$ be the Banach space of $L^1$ functions on some probability space, $\mathcal Y$ be some other Banach space, $T:\mathcal X\to \mathcal Y$ be some surjective continuous linear map, $\...
- 537
9
votes
1
answer
608
views
Interpolation theory and $C^k$-spaces
Consider the Banach spaces $C^k(M)$ ($k=0,1,2,\dots$), consisting of $k$times continuously differentiable functions $f:M\rightarrow \mathbb{C}$ on a closed manifold $M$ (or just the torus if that ...
- 779
9
votes
3
answers
684
views
Lipschitz-free spaces of $\mathbb R^n$
We define
$$
\text{Lip}_0(\mathbb R^n)=\{f:\mathbb R^n\rightarrow \mathbb R, \text{such that $f(0)=0$ and }
\sup_{x\not=y}\frac{\vert f(x)-f(y)\vert}{\vert x-y\vert}<+\infty.
\}
$$
It is well-known ...
- 16.2k
9
votes
2
answers
524
views
Projections onto $n$-codimensional subspaces of a Banach space: norms.
Hello, I'd like some help to find an answer I've been looking for since this morning.
Let $X$ be a Banach space and let $Y$ be an $n$-codimensional subspace of $X$. Let $P$ be a projection from $X$ ...
- 93
9
votes
1
answer
996
views
Topological "Interpolation" ?
Let E be a normed space, and let $T$:E * $\rightarrow$ E * be
a nonlinear operator.
Suppose that :
1) $T$ is continuous from (E *, ||.||) to itself (i.e., it is norm-continuous).
and
2) $T$ is ...
- 4,060
9
votes
1
answer
611
views
opposite Banach space
I heard this from Haskell Rosenthal many years ago.
If V is a complex vector space, say the opposite of V is the complex vector space with the same elements, the same operations except switch scalar ...
- 41.1k
9
votes
1
answer
481
views
Does the Banach space $( \ell ^2 \oplus \ell ^2 )$ have F.P.P?
The space $( \ell^2 ,\lVert \cdot \rVert _2 )$ is a Hilbert space. The space
$X=(\ell^2 \oplus \ell^2 , \lVert \cdot \rVert_\infty )$ is a Banach space. Does X have fixed point property? (For any ...
- 327
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
9
votes
1
answer
242
views
On hereditarily reflexive Banach spaces
It was proved by W.B. Johnson and H.P. Rosenthal [Studia Math. 43 (1972), 77–92]
that every Banach space $X$ with $X^{**}$ separable is hereditarily reflexive:
every infinite dimensional closed ...
- 4,461
9
votes
1
answer
462
views
Uniqueness up to isometric isomorphism of predual of $(\sum_{\lambda\in\Lambda} H_\lambda)_{l_\infty}$ where $H_\lambda$ are Hilbert spaces
This fact is an easy consequence of results of the paper Classes of Banach spaces with unique isometric preduals. by Leon Brown and Takashi Ito, but it looks like an overkill. Does anyone know a ...
- 1,697
9
votes
1
answer
893
views
Perturbations of an operator that disconnect the spectrum
The following question came to me while working on a technical matter about transversality in infinite dimension, and I'm really curious to know whether it has an affirmative answer at least under ...
- 60.5k
9
votes
1
answer
1k
views
Luxemburg norm as argument of Young's function: $\Phi\left(\lVert f \rVert_{L^{\Phi}}\right)$
Let $\Phi$ be a Youngs's function, i.e.
$$ \Phi(t) = \int_0^t \varphi(s) \,\mathrm d s$$
for some $\varphi$ satifying
$\varphi:[0,\infty)\to[0,\infty]$ is increasing
$\varphi$ is lower semi ...
- 193