All Questions
9,958 questions
3
votes
2
answers
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Is there a corresponding Hahn decomposition theorem for the real-valued Radon measures?
Hello,
As we know that a signed measure $\mu$ on $R$ can be decomposed to the positive part $\mu_+$ and negative one $\mu_-$ by the Hahn decomposition theorem.
My question is whether each real-...
- 41.1k
7
votes
2
answers
3k
views
Distributions and measures
Hello,
After reading the previous post, I still have some doubts. Let's consider everything on $R$ to avoid complications.
Can we say that any distribution $\mu\in\mathcal{D}'(R)$ of zero order is ...
CommunityBot
- 1
-1
votes
1
answer
696
views
Can singular measures be viewed as vanishing distributions? (Answer No!)
Hello,
Here is my original question: let $\mu$ be a singular measure with respect to the Lebesgue's measure on $R$. Is it true that $\int \psi \mu(d x)=0$ for any test function $\psi\in C_c^\infty(R)$...
CommunityBot
- 1
5
votes
2
answers
1k
views
Sobolev imbedding on Riemannian manifolds
Let $(M, g)$ be a non-compact smooth Riemannian manifold of dimension $n \ge 2$, and $G$ a subgroup of the isometry group of $(M,g)$, say with $G$ contained in the component of the identy.
Let $W^{1,...
CommunityBot
- 1
8
votes
3
answers
1k
views
When does a unitary Hilbert space rep of a reductive Lie group decompose into a direct sum of irreps with finite multiplicities?
I'm giving some lectures on the trace formula. Here's something I proved in the last lecture. Let $G$ be a locally compact Hausdorff unimodular topological group (e.g. a reductive Lie group), let $\...
- 23k
5
votes
3
answers
1k
views
Integration by parts for a general negative-definite self-adjoint operator.
I suspect I am asking a very stupid question.
Suppose you have self-adjoint negative-definite operator $L$ densely defined on a space $L^2(\pi)$, with $Lf = \nabla \cdot ( A(x)\nabla f)$, for some ...
- 1,471
3
votes
1
answer
1k
views
Long time behavior of the heat equation on R
Let $\mu\in\mathcal{S}'(R)$ be a Schwartz distribution. The solution of a heat equation with $\mu$ as the initial data is
$$
u(t,x)= \int_R \frac{e^{-\frac{(x-y)^2}{2t}}}{\sqrt{2\pi t}} \mu(d y)
$$
...
- 2,715
0
votes
1
answer
503
views
When are operators extended by linearity bounded?
Greetings.
Suppose that $H$ is a separable infinite-dimensional Hilbert space and that $M$ is an infinite
dimensional closed subspace of $H$. Suppose that {$v_{n}: n\ge 1$} is an infinite linearly ...
- 73.2k
9
votes
1
answer
1k
views
Distributions on product spaces
I hope this is suitable to MO.
Question. Let $X$ and $Y$ be two open sets in $\mathbb R^n$ and $\mathbb R^m$, respectively. In what sense can we consider $\mathcal{D}^{\prime} \left(X\times Y\right)$ ...
- 22.3k
9
votes
2
answers
674
views
Small crown probabilities (and infinite dimensional margin assumption)
My question is:
How do I find sharp upper bounds on $P(|q|\leq \epsilon)$ uniformly over a set of gaussian polynomes $q$ of degree two.
Notations and definitions (to make the question rigorous)
Let ...
- 96
3
votes
0
answers
498
views
PDES - from Vector fields whose inner product with their vector Laplacian equals norm of the vector field
Let $g(x_{1},........,x_{n}) = \sum_{i=1}^{n}g_{i}(x_{1},\cdots,x_{n})e_{i}$ be a function in $\mathbb{C}^n$ ($e_{i}$ are the standard bases).
Let $\nabla^{2}$ be the vector Laplacian. Let $<\cdot,...
- 800
11
votes
1
answer
603
views
Reference for a particular Radon transform on non-positively curved spaces
Let me first recall that the classical Radon transform takes a (smooth compactly supported, say) function $f$ defined on $\mathbb{R}^n$ as an input, and gives as output the map $H\mapsto \int_H f$ for ...
CommunityBot
- 1
1
vote
2
answers
177
views
Restriction on the coefficients for an operator in the free group factor $ L(\mathbb{F}_2) $
Let $\mathbb{F}_2$ denotes the free group generated by a,b, denote this group by $G$. Then consider the von Neumann algebra $L(G)$ generated by the family
$\{L_{x_g} : g \in G\}$, here, with $g \in G$...
- 5,425
0
votes
1
answer
611
views
Linear functionals and continuous functions on open intervals
Let $Q$ be an open interval of ${\mathtt R}$ and $E$ be the space of continuous and bounded functions in $Q\to \mathtt{R}$.
I call $E^*$ the set of linear functionals over $E$ and $E_+^*$ the subset ...
CommunityBot
- 1
1
vote
2
answers
2k
views
Why is the output of an LTI system the convolution of the input funtion and the impulse response?
I am looking at the description of LTI systems in the time domain.
Intuitively, I'd have guessed it would be the composition of the input function and some "system function".
$$ y(t) = f(x(t)) = (f\...
- 5,827
4
votes
1
answer
783
views
Dependence of norm of extension map on Sobolev spaces and $(\epsilon,\delta)$ domains
Let $D\subset \mathbb{R}^n$ be a bounded domain.
An extension map is $E_D: W^{p,k}(D)\to W^{p,k}(\mathbb{R}^n)$ satisfying:
...
CommunityBot
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2
votes
2
answers
2k
views
Separable quotients of non-separable Banach spaces?
I am reading the Functional Analysis book of Conway, one question from the book is find a closed subspace M of $l^{\infty}=l^{\infty}(\mathbb{N})$ with the property that $l^{\infty}/M$ is separable. I ...
- 283
-2
votes
1
answer
3k
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Separability of continuous functions with compact support [closed]
Hi,
is the space $C_0(\mathbb{R}^m)$, $m \in \mathbb{N}$ of continuous functions with compact support separable? If yes: where can I find a proof for that?
Please note: this is not a duplicate of ...
CommunityBot
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1
vote
1
answer
342
views
Compact sets in TVS
Let $K$ be a compact subset of a Hausdorff topological vector space. Is it true that
$\bigcap_{n\in \mathbb{N}}\frac{1}{n}K$ is either empty of or is a set consisting of the origin only?
- 23.3k
2
votes
2
answers
1k
views
Characterization of Weakly measurable functions
I wonder if we can characterize weak measurability of a function taking values in a Banach space using sequence of step functions (functions that have finite range) just like how we define strong ...
- 60.5k
2
votes
1
answer
208
views
Expanding Measurable Sets
Let $S,T \subset \mathbb{R}^n$ be measurable sets, and suppose that there exists a measurable bijection $f\colon S\to T$ so that
$$
\|f(x)-f(y)\| \;\geq\; \|x-y\|
$$
for all $x,y \in S$. Does it ...
- 45k
21
votes
4
answers
2k
views
A question on the integral of Hilbert valued functions
This questions stems from an attempt to recast in a form suitable for teaching some standard computations which are usually proved by handwaving, without much care about the details. My hope is that ...
- 60.5k
2
votes
1
answer
570
views
Is a polynomial positive on the sphere a sum of squares of spherical harmonic polynomials?
Let $p\in {\mathbb{R}}[x_1,\ldots, x_n]$ be a homogenous polynomial of even degree $2d$. If $p$ is positive on the unit sphere $S\subset {\mathbb{R}}^n$, then does there exist some $m>0$ and ...
CommunityBot
- 1
1
vote
1
answer
754
views
Addition of essentially self-adjoint operators
Say $A, B : H \supset D \to H$ are essentially self-adjoint operators on the dense common domain $D$. $H$ is some Hilbert space. Does it hold that $A + B$ is also essentially self-adjoint? If not, can ...
- 13k
1
vote
0
answers
369
views
Infinite internal direct sums of subspaces
Given a compact Hausdorff space $K$ such that $C(K)$ is of density $\omega_1$. Suppose that every copy of $c_0(\omega_1)$ in $C(K)$ is complemented. Let $\{Y_\alpha\colon\alpha<\omega_1\}$ be a ...
- 11.3k
0
votes
2
answers
444
views
Sobolev space: probably simple ode....
I am trying to solve for $y(x)$ in terms of $f(x)$ in a convenient space (eg. $\dot{H}^2(\mathbb{T})$-zero mean). Here is the ode:
$y(x)+y(x)y'(x)=f(x)$.
I think a contraction mapping argument will ...
- 1
2
votes
0
answers
140
views
WLD Banach spaces
Does anyone know of an example of a weakly Lindeloff determined (WLD) Banach space which does not contain c_0 and is not weak Asplund? I believe the example of a WLD, non-weak Asplund space by Argyros ...
- 21
7
votes
3
answers
4k
views
Measures on infinite dimensional Banach spaces
Does there exist a Borel measure or any valid measure on an infinite dimensional Banach space such that a bounded open set in this space has a positive measure ?
1
vote
2
answers
515
views
continuity of extension of maps along curves
Let $a\le b$ and $k\ge 0$ be given and fixed. Let furthermore $x$ and $y$ denote two different elements of a Hilbert space $H$. Suppose $u:\mathbb{R}\rightarrow H$ is a $C^k$-embedding connecting $x$ ...
CommunityBot
- 1
3
votes
2
answers
1k
views
Do the Euler method's approximations always approach the true solution?
Let $B$ be a Banach space and $f : [0,+\infty)\times B \to B$ be a continuous function which is Lipschitz continuous in the second argument with Lipschitz constant $L$ (which does not depend on the ...
- 1,171
2
votes
0
answers
366
views
Are affine continuous functions on Bauer sub-simplices of the probability measures given by integration over continuous functions?
Let $X$ be a compact (non-metrizable) Hausdorff space and $\mathcal{P}(X)$ the set of Radon probability measures with
weak-$*$ topology (weak topology induced by the continuous functions).
Consider a ...
- 888
9
votes
2
answers
477
views
An extension of Gaussian Isoperimetry
The Gaussian isoperimetric inequality (Tsirelson,Sudakov, Borell) states that among all sets of given Gaussian measure in the n-dimensional Euclidean space, half-spaces have the minimal Gaussian ...
- 6,694
7
votes
1
answer
592
views
topologies on U(H)
There are many topologies on the algebra $B(H)$ of bounded operators on Hilbert space:
the weak, strong, ultraweak (also called σ-weak), ultrastrong (also called σ-strong), and some more......
- 11.1k
0
votes
1
answer
622
views
products in the category of banach spaces
Let $\{X_{\alpha} \}_{\alpha \in A}$ be a collection of Banach spaces. It is easy to show that $ P = \{(x_{\alpha}) : {\rm sup}_{\alpha} \|x_{\alpha} \| < \infty \} $ with $\| (x_{\alpha} ) \| = {\...
- 25.8k
3
votes
4
answers
2k
views
Suitable references for the the Stone-von Neumann Theorem
Hi all,
I am working on a mathematical physics project now and I need to understand the Stone-von Neumann Theorem properly. Wikipedia says that it is any one of a number of different formulations of ...
CommunityBot
- 1
7
votes
1
answer
423
views
Best constant in comparison between Rademacher and gaussian averages?
Let $(g_k)$ be a sequence of independent standard gaussians variables on a fixed probability space $\Omega$. Let $(\epsilon_k)$ be a sequence of independent rademacher variables.
What is the best ...
- 31.5k
3
votes
0
answers
637
views
Fixed point theorem for convex, closed multivalued mapping
There is well-known fixed point theorem theorem for multivalued l.s.c. maps, based on Michael selection theorem:
Suppose, that $X$ is compact, convex and metrizable in locally convex Hausdorff ...
- 696
12
votes
4
answers
4k
views
Locally constant functions with compact support = smooth ?
Hello,
I have a trivial question, but I hope that you don't mind helping. I often get confused with basic definitions.
Let F be a p-adic field. Then (from what I understand) $C_c^{\infty}(F)$ is the ...
- 3,584
5
votes
1
answer
878
views
Short five lemma in Banach spaces
Denote by $\mathbf{Ban}$ the category of Banach spaces and bounded linear maps and by $\mathbf{Banc}$ the subcategory of Banach spaces and linear contractions. The isomorphisms of $\mathbf{Ban}$ are ...
- 41.1k
6
votes
1
answer
805
views
Ergodicity of Convoluted White Noise
I have a question regarding ergodicity in infinite dimensional spaces.
Let $\mathcal{D}$ be the space of distributions on a Schwartz space, and let $\mu$ be the white noise process which exists by ...
5
votes
2
answers
579
views
Improved versions of discontinuous functions
Given a set X (such as the set of points in an interval), the space ℝX of all real-valued functions on X is not usually the function space we work with -- it is "too large" in some sense. Thus, ...
- 13k
2
votes
1
answer
265
views
Multiple ergodic averages with varying number of terms
Hi. I've been stuck on the following question for some time.
Consider a sequence of functions $\left( f_n \right)$ from an ergodic space $\left( \mathsf{X}, \mathsf{S}, \mu \right)$ to $\left[ 0,1 \...
- 17k
9
votes
2
answers
928
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Property (T) for pseudogroups
Let $H$ be a Hilbert space, $S(H)$ be the inverse semigroup (pseudogroup) of linear maps between (closed) subspaces of $H$ preserving the dot product (the operation is composition of partial maps). ...
CommunityBot
- 1
9
votes
1
answer
2k
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The Invariant Subspace Problem: examples
Question. Is there a concrete example of a bounded linear operator on a Hilbert space for which it is not known if it has a non-trivial closed invariant subspace?
[Added 24.01.2011: According to ...
- 512
0
votes
1
answer
365
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Integral in a σ−convex set.
Having had no (proper) answer to this question, I formulate the remaining case as a new question as follows. With $I=[0,1]$, let $E$ be a separable (real) Banach space, and let $\gamma:I\to E$ be ...
CommunityBot
- 1
14
votes
3
answers
3k
views
The difference between $l^1(G)$ and the reduced group $C^*$ algebra $C_r^*(G)$
Let $G$ be a group and $l^2(G)$ the Hilbert space on $G$. The complex group algebra $CG$ can be imbedded in $B(l^2(G))$, the set of all bounded linear operators, by left translation. The reduced group ...
- 11.1k
4
votes
3
answers
1k
views
Set of invertible operators in B(H) is connected. Is it true? Is there a reference?
Suppose $H$ is a Hilbert space, $B(H)$ is the algebra of bounded linear operators on it, $K(H)$ is ideal of compact operators in $B(H)$, $Inv(B(H)/K(H))$ is the topological group of invertible ...
- 11.1k
2
votes
1
answer
431
views
Sobolev imbedding
It is known that, for $n \ge 3, 2 < p< 2^*$, the imbedding $H^1(\mathbb{R}^n) \hookrightarrow L^p(\mathbb{R}^n)$ is not compact. Let $G=O(n_1) \times O(n_2)\times\cdots\times O(n_k)$, with
$n_1+...
- 397
8
votes
1
answer
1k
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Ring of continuous functions, reference request.
I am looking for a reference for the following facts in functional analysis and topology. (If these "facts" are not true, I suppose I'm looking for the closest approximation which is true.)
Let $X$ ...
- 13.3k
23
votes
1
answer
2k
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Which Fréchet spaces have a dual that is a Fréchet space?
I've read the claim that Fréchet spaces that are not Banach spaces never have a dual that is a Fréchet space, but have not been able to find a proof of this statement. Is it trivial or does someone ...
- 22.3k