All Questions
1,222 questions
8
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On the classification of injective Banach spaces
A Banach space $E$ is injective when it is complemented in each Banach space $X$ that contains it as a closed subspace. The space $E$ is $1$-injective if the copy of $E$ in $X$ is the range of a norm-...
- 4,461
9
votes
3
answers
868
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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=...
- 31.5k
10
votes
1
answer
591
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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 ...
- 31.5k
2
votes
3
answers
757
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The importance of basis constant in Banach spaces
Let $X$ be a Banach space an let $(e_n)_{n=1}^{\infty}$ be a Schauder basis of $X$. If we denote the sequence of the natural projections associated with $(e_n)_{n=1}^{\infty}$ by $(S_n)_{n=1}^{\infty}$...
- 31.5k
12
votes
1
answer
306
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Containment of $c_0$
I have the following question. I guess it's quite simple for experts.
Unfortunately, I could not come up with an answer yet.
Let $X$ be a Banach space which contains no copy of $c_0$.
Does it impply ...
- 31.5k
2
votes
0
answers
108
views
Suprema and infima in spaces ordered by non-normal cones
Background
We shall call a subset $V_+ \subseteq V$ of a Banach space $V$ a cone if
$V_+$ is closed,
$\alpha V_+ \subseteq V_+$ for all $\alpha \geqslant 0$, and
$V_+ \cap (-V_+) = \{0\}$.
Cones ...
- 123
15
votes
1
answer
2k
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Quotients of $\ell_\infty$ by separable subspaces
Given a (closed) separable subspace $M$ of $\ell_\infty$, I am interested in conditions implying that the quotient $\ell_\infty/M$ is isomorphic to a subspace of $\ell_\infty$.
It is not difficult ...
- 31.5k
3
votes
2
answers
423
views
Banach algebra for measures induced by Haar measures
It is classical that $L^1(G, m)$ is a Banach algebra when $G$ is a locally compact group with Haar measure $m$ by using the operation of convolution via the integral
$$(f*g)(y)=\int_Xf(x)g(yx^{-1})\,...
- 11.2k
12
votes
2
answers
3k
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Direct proof of injectivity of $L_\infty$
I would like to know a simple proof of isometric injectivity of $L_\infty$. The proof I've found in Topics in Banach space theory. F. Albiac, N. Kalton uses two deep result.
$L_\infty$ as ...
- 1,697
1
vote
0
answers
52
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Extension of $S_+$ type operators
Let $X$ be a reflexive Banach space and $G\subset X$ a open bounded set. Let $F:\overline{G}\rightarrow X^\star$ be a $S_+$ operator, i.e., if for any sequence $x_n$ in $G$ for which $x_n\...
- 6,210
2
votes
1
answer
4k
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Precise versions of "differential operators are unbounded but closed linear operators"
I am trying to understand to what extent the following result of Hille is an extension of the usual theorems on differentiation under the integral sign.
Theorem (Hille). Let $(\Omega,\Sigma,\mu)$ ...
CommunityBot
- 1
6
votes
1
answer
275
views
Quasi-reflexive spaces which are not isometric to dual spaces
My question may sound weird and I have no deep motivation behind it other than curiosity.
As is well-known, quasi-reflexive spaces have the Radon-Nikodym property hence their balls have lots of ...
- 31.5k
7
votes
0
answers
293
views
Complex interpolation of a Banach space and its antidual when the space has a basis
Given a complex Banach space $X$ and its antidual $\hat X^*$, it is possible in some cases to apply the complex interpolation method, and get as $(X,\hat X^*)_{1/2}$ a Hilbert space. See [F. Watbled, ...
- 4,461
8
votes
3
answers
1k
views
Dual Banach space of $B(X,Y)$ when $X$ is finite dimensional
Denote $B(X,Y)$ the Banach space of bounded operators between Banach spaces $X$ and $Y$.
When $X$ and $Y$ are both finite dimensional, it follows from the formula
$$\|u\|_{B(X,Y)} = \sup_{\|x\|_X <...
- 31.5k
0
votes
0
answers
362
views
Banach-Mazur distance estimate finite-dimensional $\ell_p$ spaces
Hey my fellow Banach space guys. Sorry for another elementary question, and yes I have looked for the past hour and a half to see if I can find it on Google or in my books at home.
Fix $n\in\mathbb{...
- 3,993
7
votes
1
answer
556
views
Non continuous Linear form on $E=C([0,1],\mathbb{R})$ without AC
Let's note $E=C([0,1],\mathbb{R})$ the Banach space of real continuous funtions from the [0,1] interval with the uniform norm.
Is it possible to show a non-continuous linear form on $E$ exists ...
- 39.8k
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
8
votes
0
answers
357
views
Ultrapowers of Banach spaces without the continuum hypothesis
Let $\mathcal{U}$ be a non-trivial ultrafilter on the set of integers $\mathbb{N}$, and let $C(K)$ denote the Banach space of continuous functions on a compact $K$. Under the continuum hypothesis CH, ...
- 4,461
2
votes
0
answers
172
views
Projective tensor powers of Banach spaces over a normed field
Let $E$ be a Banach space over a complete normed field $\mathbb K$. Is it possible to classify all proper $E$ for which the projective seminorm $p_n$ defined on the $n$-th tensor power of $E$ is a ...
- 371
6
votes
1
answer
353
views
Sequential continuity of linear operators
Let $u\colon L\to M$ be a linear map of locally convex linear topological vector spaces.
Assume that $u$ is sequentually continuous, i.e. maps convergent sequences to convergent ones.
(This notion is ...
- 7,500
1
vote
0
answers
295
views
Is reflexive Banach space valued scalarwise Lebesgue space isomorphic to the Bochner space?
I first specify the setting and then formulate the question precisely. (A very long post follows.)
Definitions 1. For $E$ a (real Hausdorff) locally convex space, say that $E$ is suitable iff there ...
CommunityBot
- 1
2
votes
0
answers
648
views
Closed-form expressions for dual norms of real normed vector spaces
Didn't get any biters over at MSE, so I figure this place might be more appropriate...
Say that $V$ is a finite-dimensional real normed vector space, where for some $v \in V$ the norm is notated by $\...
- 4,907
2
votes
0
answers
117
views
Maximum Principle with Banach Control Space
This is a problem that seems very natural to me, but I couldn't find any formal statement in the literature for some time now. I am basically considering an autonomous optimal control problem in which,...
- 197
3
votes
1
answer
245
views
Orthonormal basis in $\ell^n_p$
Given a $k$-dimensional subspace in $\ell^n_p$, is there a way to bound the value of
$$
\sum_{i=1}^k \|a_i\|_{\ell^p}^2
$$
for $a_i$ an orthonormal (for the "standard" underlying $\ell^n_2$) basis.
...
- 4,432
5
votes
1
answer
578
views
Infimum over all vector-valued L^2 spaces
Suppose I have a Banach space $E$ (which may be finite dimensional if you wish), a Hilbert space $H$ and a tensor $\tau \in H\otimes E$ in the algebraic tensor product. There are lots of ways to ...
- 60.5k
2
votes
1
answer
373
views
Is it true that $c_0(X)^* = \ell_1(X^*)$ ?
I'm trying to prove this that but I can't . Any help/reference ?
- 60.5k
11
votes
1
answer
964
views
Quotients of l^infty
Let $M$ be a closed subspace of $l^\infty$. Suppose that the quotient $l^{\infty}/M$ is isomorphic to $l^\infty$. Is it true that $M$ is complemented in $l^\infty$?
- 31.5k
4
votes
1
answer
280
views
Approximation of an integral over the unit ball of L_1
For every $\varepsilon>0$ find a piecewise continuous function $q:[0,1]\rightarrow \mathbb{R}$ such that $\int_0^1 q(x)dx=1$ and
$$\int_{0}^1 \int_{0}^{s} \left|\frac{q(s)q(t/s)}{s}- \frac{q(t)q((s-...
- 9,486
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 ...
CommunityBot
- 1
4
votes
0
answers
90
views
$x\in Ext(B_X)$ has the Kadec property, implies that the slices form a neighborhood base of the norm topology
This is question 3.87 from Fabian's Functional Analysis and Infinite-Dimensional Geometry. The result is credited to Lin and Troyanski. Where on the net can I read a proof of this lemma? Any help ...
5
votes
1
answer
398
views
What Approximation Property does the space of Schatten-p class operators have?
Background
This is a follow-up question to:
What (classes of) Banach spaces are known to have Schauder basis?
In the previous question, I asked about what spaces are known to have Schauder basis. It ...
CommunityBot
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7
votes
2
answers
657
views
Subspaces isomorphic to $C[0, \omega_1]$
Let $\omega_1$ be smallest uncountable ordinal. I am trying to understand the possible "large" subspaces of $C[0,\omega_1]$, namely those which are isomorphic to the whole space. Therefore I have the ...
CommunityBot
- 1
3
votes
2
answers
362
views
Invariant subspaces for compact restrictions
Suppose $Y$ is a closed hyperplane in $X$, so we can write $X=Y\oplus[x_0]$. Let $y_n$ be a normalized basis of $Y$. Define an operator $S:Y\to Y\oplus[x_0]$ by $Sy_n=\alpha_ny_n+\beta_nx_0$, for any $...
- 4,729
1
vote
0
answers
126
views
Is scalarwise measurability determined by the strong dual?
Since this question has not received an answer so far, I try to reformulate the question in a simpler manner as follows: Do there exist $E,F,\ell,f$ such that
$E$ and $F$ are separable (real) Banach ...
CommunityBot
- 1
15
votes
5
answers
680
views
Idiosyncratic characterizations of $\ell^p$, for $p\not=1,2,\infty$
Do there exist, either in the literature or in folklore, theorems
that characterize some particular $\ell^p$ space(s) ($p\not=1,2,\infty$)?
Such a theorem should reveal the particular space(s) as ...
- 22.3k
16
votes
1
answer
2k
views
What (classes of) Banach spaces are known to have Schauder basis?
Motivation:
I am trying to see for what class of Banach spaces the following result is true:
There exists an increasing sequence of finite dimensional subspace {$V_n$} of a Banach space X (with ...
- 31.5k
1
vote
1
answer
353
views
Agreement of two topologies on a linear space
I'm dealing with the formalism of an abstract Wiener space, and I'm not sure if two relevant topologies coincide.
Let $X$ be a topological vector space, and let $X^*$ be its dual space of continuous ...
- 16.4k
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") ...
- 23k
7
votes
1
answer
439
views
series representation in injective tensor products
All books on tensor products of Banach spaces contain the well-known theorem of Grothendieck that every element of the completed projective tensor product
$X \tilde{\otimes}_ \pi Y$ has a ...
CommunityBot
- 1
3
votes
1
answer
389
views
Continuity of lattice operations in Banach lattices
Let $L$ be a Dedekind-complete Banach lattice. Let $\mathcal{B}$ be the family of nonempty norm-compact subsets of $L$ that are bounded from below. Endow $\mathcal{B}$ with the topology induced by ...
- 31.5k
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
1
vote
0
answers
393
views
Unambiguous "weak" vector valued $L^{+\infty}$ spaces?
For some time, I have been stuck to the problem to be described as follows. The (perhaps not so commonly known) facts given here are taken from R. E. Edwards' Functional Analysis (Holt, Rinehart and ...
CommunityBot
- 1
1
vote
1
answer
184
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Special kind of operators
Consider an operator $A: H \longrightarrow X$ ($H$ is a Hilbert space and $X$ is a Banach space) that has a representation
$$ A = \sum_{j=0}^\infty a_j \langle \cdot, e_j\rangle \cdot x_j,$$
where $(...
- 25.3k
6
votes
1
answer
249
views
What is the doubling dimension of convex functions?
I am interested in the complexity of convex functions, specifically the "doubling dimension" of the class of convex functions defined on a compact subset of Euclidean space, when compared using the $L^...
- 45k
4
votes
2
answers
535
views
On hyperplanes of $L\infty$
Consider the hyperplane $H=\{f\in L^\infty: \int f = 0\}$ of $L^\infty = L^\infty[0,1]$. My question is:
1. What is the Banach-Mazur distance between $H$ and $L^\infty$? Are there "natural" ...
- 607
2
votes
0
answers
301
views
Finite codimensional subspaces of L(X,Y)
Let $X$ and $Y$ be separable Banach spaces and $L(X,Y) $ be the Banach space of bounded linear operators from $X$ to $Y$. Suppose $A$ is a norm closed finite codimensional subspace of $L(X,Y)$.
My ...
- 1,073
3
votes
0
answers
642
views
question about bidual normed space
Consider a Banach $\mathbb{R}$-space E and an element $u\in E''$ such that :
for all sequence $\phi_n\in E'$ which $\sigma(E',E)$-converges to $\phi \in E'$, one has $\lim u(\phi_n)=u(\phi)$.
Is it ...
CommunityBot
- 1
7
votes
2
answers
1k
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Weak*-closed and complemented subspaces of dual Banach spaces
We consider a Banach space $X$ and its dual $X^*$.
Let $Q\colon X^\ast \to X^\ast$ be an idempotent operator.
Question:
Can we find an idempotent operator $P\colon X^\ast \to X^\ast$ which is weak$...
- 2,378
11
votes
2
answers
2k
views
Complexifying a real Banach space and its dual
A standard way to define the "complexification" $E_\mathbb{C}$ of a real Banach space $E$ is to define a complex linear structure on $E\times E$ by (1) $(x,y)+(u,v)=(x+u, y+v)$, (2) $(a+ib)(x,y)=(ax-...
- 1,704
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