All Questions
Tagged with fa.functional-analysis banach-spaces
1,222 questions
0
votes
1
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128
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Regarding an element being self adjoint
Let $A$ be a unital C*-algebra. Let $x,y\in A$ be self adjoint elements in $A$, with $x$ being invertible. Can we say that the spectrum of $x^{-1}y$ is a subset of the real line? I know this is true ...
- 12.5k
4
votes
2
answers
2k
views
Convergence of Gaussian measures
Let $X$ be a separable Banach space with its Borel $\sigma$-algebra $\mathcal F$. Let $x_n \to x$ in $X$. Fix a Gaussian covariance operator $K$, and let $\mathbb P_n$ and $\mathbb P$ be Gaussian ...
- 1
6
votes
3
answers
2k
views
Space of compact operators
I am interested in the Banach space $\mathcal{K}=\mathcal{K}(\ell^2)$ of compact operators on $\ell^2$, however my questions can be stated for any $\mathcal{K}(E)$, where $E$ is an arbitrary Banach ...
- 2,605
4
votes
1
answer
330
views
Uniform boundedness principle for almost surely converging sequence of operators
I'd like to do the following: I consider a separable Banach space $X$ with a probability measure $\mu$ on the Borel $\sigma$-algebra $\mathcal B(X)$. Additionally, I have a sequence of measurable, ...
- 29.8k
3
votes
1
answer
105
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G.L. l. u. st. for subspaces of Banach spaces with an unconditional basis
A Banach space $X$ has Gordon-Lewis local unconditional structure (G.L. l. u. st.) if for every finite dimensional subspace $E$ of $X$, the inclusion operator $i:E\to X$ factors through a finite ...
- 31.5k
2
votes
1
answer
349
views
$K$-convex Banach spaces
Let $X$ be a Banach space. We say that $X$ contains $\ell_1^n$'s uniformly iff for all $n\in\mathbb N$ there exist subspaces $X_n\subseteq X$ with $d(X_n,\ell_1^n)\leq \lambda$ for some $\lambda\geq 1$...
- 1,879
1
vote
0
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124
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For which Banach spaces is the self composition operator Lipschitz?
Let $X\subseteq \{f|f:D\rightarrow \mathbb{R}^n\}$ be a Banach space, with at least all polynomials on $D$ contained in $X$, where $D\subseteq \mathbb{R}^n$ is open and bounded.
Let $U\subseteq X\cap \...
- 183
2
votes
0
answers
89
views
Another question about asymptotic models in Banach spaces
The array $(x_{i}^{k})_{i=1,k\in\mathbb{N}}^{\infty}$ of normalized $M$-basic sequences in a Banach space $X$ is itself called $M$-basic if, for every $k\leq i_{1}<i_{2}<\ldots$, the diagonal ...
- 201
3
votes
0
answers
274
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Density of signed measures in dual space
Let $B$ be a Banach space of functions on a Radon space $X$. By the Hahn-Banach theorem, we know that the canonical evaluation map is isometric. That is, for every $f \in B$, we have
$$\|f\| = \sup_{\...
- 103
0
votes
0
answers
67
views
Dual of isometric copies into dual Banach spaces
Let $X$ be a Banach space and $X_1\xrightarrow{}X$ isometrically. Under some assumption can we guarantee that $X^*$ contains an isometric copy of $X_1^*$. I am also interested to know if this happens ...
- 63.9k
0
votes
0
answers
129
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Certain decompositions of decomposable Banach spaces
Let $\mathcal{X}$ be a decomposable Banach space (i.e. a topological direct sum of infinite-dimensional subspaces, say $\mathcal{X}=\mathcal{A}\oplus\mathcal{B}$). Can one always obtain another ...
- 1,453
3
votes
0
answers
138
views
Property $(V_1)$ for Banach spaces
This aim of this note is to record a problem that still seems to be open.
Räbiger, in his doctoral thesis, defined property $(V_1)$ as follows: A Banach space $X$ has property $(V_1)$ if every ...
- 2,605
6
votes
2
answers
509
views
A question on Grothendieck space
A Banach space $X$ is said to be Grothendieck if the weak and the weak* convergence of sequences in $X^{*}$ coincide. I have the following two questions.
Question 1. A Banach space $X$ is Grothendieck ...
- 2,605
4
votes
1
answer
196
views
On the intersection of two Orlicz spaces
It is well-known that if $1\leq p\leq q\leq \infty $ then
$$ L^p(X)\cap L^q(X)\subset L^r(X)\quad\quad \text{whenever $r\in [p,q]$}\tag{I}\label{Eq}.$$
Indeed let $u\in L^p(X)\cap L^q(X)$. For some $...
- 1,869
4
votes
1
answer
396
views
Closedness of the image of the unit ball
Let $X$ be a Banach space and let $P$ be a bounded, linear projection on $X$. Is $P[B_X]$ closed in $X$? Here $B_X$ is the closed unit ball of $X$.
This is trivial if $X$ is reflexive, but otherwise ...
- 42.8k
4
votes
1
answer
366
views
Example of empty projection in strictly convex Banach space
Let $X$ be a strictly convex Banach space, let $C\subseteq X$ be a nonempty closed convex set, and let $P_C$ be the set-valued metric projection
$$P_C(x) = \{y\in C : \|x-y\| = d(x,C)\} . $$
We know ...
- 60.5k
2
votes
1
answer
280
views
Finite-dimensional subspaces of $c_{0}$
Let $M$ be a finite-dimensional subspace of $c_{0}$, and let $\varepsilon>0$.
Question. Does there exist a finite rank projection from $c_{0}$, of norm $\leq 1+\varepsilon$, onto a subspace $N$ of ...
- 66.3k
3
votes
1
answer
451
views
Uniform smoothness and twice-differentiability of norms
To get to the simplest case, consider a norm $\|\cdot\|$ over $R^n$ that is uniformly smooth of power-type 2, that is, there is a constant $C$ such that $$\frac{\|x+y\| + \|x - y\|}{2} \le 1 + C \|y\|^...
- 408
10
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0
answers
226
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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
6
votes
0
answers
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
1
vote
1
answer
130
views
Both $\ell_\infty$ and $L_\infty$ belong to $\mathcal{B}_1$ [duplicate]
I am looking for a resource or some hints on why the two normed spaces $\ell_\infty$ and $L_\infty$ belong to the family $\mathcal{B}_1$, that is,
they are of the family of Banach spaces $X$ such that ...
- 42.8k
1
vote
0
answers
68
views
Inequality of exponentials of Banach operators
(I have moved this question from Stackexchange).
Given the operators $\{A_j\}$ in a Banach algebra and a positive integer $p$, let
\begin{equation}
g=\exp\left(\frac{1}{n}\sum_{j=1}^p A_j\right)\quad\...
- 111
2
votes
1
answer
143
views
How to characterize the order convergence in Bochner-integrable functions space?
Let $(\Omega,\Sigma,\mu)$ a finite measure space. We want to characterize the order convergence (for sequences) in Bochner integrable functions space $L^1(\mu,X)$, $X$ Banach lattice.
In $L^p$ we have:...
- 21
5
votes
1
answer
724
views
Embedding of a Banach space into a Hilbert space
Let $\mathbb H$ be a Hilbert space and let $\mathbb B$ be a Banach space continuously embedded in $\mathbb H$ and distinct from $\mathbb H$. Is it true in general that $\mathbb B$ is an $F_\sigma$ of ...
- 16.4k
3
votes
1
answer
178
views
Banach Mazur distance between the cube and the cross-polytope in the dimensions for which a Hadamard matrix exists
The Banach-Mazur distance between two centrally symmetric convex bodies $K,L\in\mathbb{R}^n$ can be defined as
$$ d(K,L) = \inf \{ r : \exists T\colon \mathbb{R}^n \to \mathbb{R}^n \text{ linear such ...
- 183
2
votes
0
answers
57
views
Is this Beppo-Levi curl space a Banach space?
Let us define the quotient space:
$$ V = \{ \mathbf{u} \in L^2_{loc}(\mathbb{R}^3; \mathbb{R}^3) : \operatorname{curl} \mathbf u \in L^2(\mathbb{R}^3; \mathbb{R}^3) \} / \nabla H^1_{loc}(\mathbb{R}^3)....
- 163
1
vote
1
answer
144
views
What's the size of non standard monad for weak topology?
There have been several works characterizing weak topology by nonstandard analysis, which give rise to the following monad ($X$ is a Hilbert space):
$$\mu(0) = \{y\in{}^{*}X: \forall x\in X ~~ \...
- 41.1k
3
votes
1
answer
128
views
The weak*-convergence of the summing basis of $c_{0}$
Suppose that $(x_{n})_{n}$ is a sequence in a Banach space $X$. We let $\textrm{clust}_{X^{**}}((x_{n})_{n})$ be collection of all the weak*-limit points of $(x_{n})_{n}$ in $X^{**}$.
Let $(e_{n})_{n}$...
- 18.7k
2
votes
1
answer
247
views
Is the union of good equivalence relations on a compact space good?
Let $X$, $Y_1$ and $Y_2$ be a compact Hausdorff spaces and let $\varphi_i:X\to Y_i$ be a continuous surjection (and so a quotient map).
Let $\sim$ be the minimal closed equivalence relation on $X$ ...
- 41.8k
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 ...
- 66.3k
2
votes
3
answers
580
views
How do I apply Brouwer fixed-point theorem in this claim?
Let $\zeta:\mathbb{R}\to [0,+\infty)$ be a continuous non-negative function such that $\zeta(0)=0$ and $\tau\mapsto \zeta(\tau)\tau$ is a non-decreasing differentiable function whose derivative is ...
- 1,101
30
votes
1
answer
1k
views
Functional-analytic proof of the existence of non-symmetric random variables with vanishing odd moments
It is known that a random variable $X$ which is symmetric about $0$ (i.e $X$ and $-X$ have the same distribution) must have all its odd moments (when they exist!) equal to zero. The converse is a ...
- 6,853
1
vote
1
answer
171
views
$l_{1}$-block basic sequences in Banach spaces with an unconditional basis
Let $X$ be a Banach space with an unconditional basis $(x_{n})_{n}$.
Question. If $X$ contains a subspace isomorphic to $l_{1}$, does $(x_{n})_{n}$ admit a block basic sequence equivalent to the unit ...
- 3,065
4
votes
1
answer
566
views
Fréchet vs. Carathéodory differentiability on Banach spaces
It is well-known that in the finite-dimensional case one can use the notion of Fréchet differentiability and Carathéodory differentiability interchangeably. See for example the 194 AMM article Frechet ...
- 548
3
votes
0
answers
245
views
Regularity of the dependence of the flow on the vector field definining it
Let $M$ be a smooth compact manifold and $k \geqslant 1$.
Define $\mathfrak{X}^k(M,TM)$ to be the set of vector fields $M \rightarrow TM$ of class $C^k$. As $M$ is compact, endowing $\mathfrak{X}^k(M,...
15
votes
3
answers
2k
views
Disintegrations are measurable measures - when are they continuous?
This is a sequel to another question I have asked.
The notion of disintegration is a refinement of conditional probability to spaces which have more structure than abstract probability spaces; ...
- 5,405
0
votes
0
answers
168
views
Sequence of functions tending to zero in L^2
Let us consider a sequence of functions $f_n : (0,1)\times (0,1) \to \mathbb{R}$ in $L^2((0,1)\times (0,1))$ satisfying the following condition:
$$
\lim_{n \rightarrow \infty}\int_{1/j}^{1 - 1/j}\...
- 14.2k
2
votes
0
answers
62
views
Decomposition of the Orlicz norm into sequential norm
I am bearing seeking for a sequential decomposition of the norm in Orlicz space.
Let me state what is known in the particular case of Lebesgue space $L^p(\Bbb R^d)$.
Given $u\in L^p(\Bbb R^d)$ let
$$n\...
- 1,101
2
votes
1
answer
133
views
Control on dimension of image
Let $f:E\rightarrow F$ be a map between Banach spaces E and F; E finite dimensional (>0) and F infinite dimensional. Let $F$ be equipped with its weak topology and suppose that $f$ is strong-weak ...
- 8,098
3
votes
1
answer
951
views
Specific criterion for the sum of two closed sets to be closed
Let $Y$ and $Z$ be two closed subspaces of a Banach space $X$ with $Y\cap Z=\{0\}$.
I know that $Y+Z$ is a closed subspace of $X$ $\iff \exists \alpha > 0:\quad \lVert y\rVert \le \alpha\lVert y+z\...
- 12.9k
5
votes
2
answers
247
views
Is there a topology that makes every basic sequence null?
Let $E$ be a Banach space. Let $F$ be the collection of all $f\in E^*$ such that $\left<f,e_n\right>\to 0$, for every normalized basic sequence $\{e_n\}$. It is easy to see that $F$ is a closed ...
- 5,529
34
votes
8
answers
9k
views
When is a Banach space a Hilbert space?
Let $\mathcal{X}$ be a real or complex Banach space.
It is a well known fact that $\mathcal{X}$ is a Hilbert space (i.e. the norm comes from an inner product) if the parallelogram identity holds.
...
- 5,670
4
votes
2
answers
419
views
$C[0,1]$ is not a Grothendieck space
A Banach space $X$ is called a Grothendieck space if $\text{weak}^{*}$-null sequences in $X^{*}$ are weakly null. Some of the classical Grothendieck spaces are the $C(\Omega)$ spaces if $\Omega$ is ...
- 3,065
0
votes
0
answers
302
views
Convergence of characteristic functions vs. weak convergence of measures and the Ito-Nisio theorem
In section 2.6 of Linde's Probability in Banach Spaces: Stable and Infinitely Divisible Distributions the author is pointing out that in infinite-dimensional Banach spaces the convergence of ...
- 167
0
votes
1
answer
137
views
Lower semi-continuity of induced function on sequences
Let $f:X\rightarrow [0,\infty)$ be (resp. weakly) lower semi-continuous on the reflexive Banach space $X$. Let $\ell^p(X)$ denote the space of $p$-summable sequences in $X$, i.e.: $\sum_{n=1}^{\infty}...
- 2,101
12
votes
1
answer
467
views
Subtracting the weak limit reduces the norm in the limit
Question
Let $X$ be some reflexive Banach space. Suppose $x_n$ is some sequence in $X$ that weak converges to some $y \neq 0$. Is it the case that
$$ \limsup \|x_n - y\| < \limsup \|x_n\| ?$$
...
- 1,453
6
votes
2
answers
888
views
Weak convergence in the intersection of Lebesgue spaces or Sobolev spaces
Let $B:=B_1\cap B_2\cap...\cap B_n$, where each $B_j$ is a reflexive Lebesgue space or Sobolev space (such as $L^4$, $H^1$, etc.) on a domain in $\mathbb{R}^d$. Then $B$ is a Banach space endowed with ...
1
vote
0
answers
64
views
Embedding a normed space as a hyperplane
Let $X$ be a real normed space and suppose that $X$ is a closed hyperplane of a bigger space $\tilde X$. Given any unit vector $u$ in $\tilde X\setminus X$, consider the function $p:X\to\mathbb R$ ...
- 483
1
vote
0
answers
45
views
Generalizations of the Wiener Tauberian Theorem to Musielak-Orlicz spaces
Musielak-Orlicz spaces provide a generalization of the usual $L^p$ spaces on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ to spaces of functions for which the Luxemburg norm
$$
\|f\|_M:=\inf\left\{\lambda &...
- 5,405
3
votes
0
answers
145
views
Non uniqueness of center of the Banach-Mazur compactum
In "The Banach-Mazur distance to the cube and the Dvoretzky-Rogers factorization" Szarek and Bourgain prove a proportional Dvoretzky-Rogers factorization :
Given $1>\delta>0$ , there ...
- 245