All Questions
10,051 questions
9
votes
1
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Alternative proof of a theorem of Riesz
My question is not research level, but I have not received any feedback on Mathstack; so I am posting it here. I am aware of the traditional proof of the Riesz Theorem that relates linear functionals ...
- 263
9
votes
1
answer
698
views
Is this Hankel matrix in trace class
Let A be the infinite Hankel matrix with the coefficient
$$A_{kj}=e^{(-t(k+j)^2)}-e^{(-t(k+j+2)^2)},$$ with $t$ a nonnegative real number.
Is $A$ in trace class with a norm bounded by an absolute ...
- 93
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
596
views
Classical analogue of the Stone-von Neumann Theorem?
Let $U_s$, $V_t$ be a pair of continuous $n$-parameter groups ($n < \infty$) of unitary operators on a complex Hilbert space $\mathcal{H}$. The Stone-von Neumann Theorem establishes that any such ...
- 657
9
votes
1
answer
456
views
Embeddings of Sobolev-Orlicz spaces
The Birnbaum--Orlicz spaces generalize the Lebesgue spaces (see http://en.wikipedia.org/wiki/Birnbaum-Orlicz_space for a precise definition). The space $L_\Phi(\Omega)$ is defined for convex functions ...
- 52.3k
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 ...
- 60.6k
9
votes
1
answer
511
views
Do these surfaces intersect?
For any real numbers $a_{1},a_{2},\cdots a_{6}$ and $b_{1},b_{2},\cdots b_{6}$
with $\sum_{i=1}^{6}a_{i}^{2}=1$ and $\sum_{i=1}^{6}b_{i}^{2}=1$,
does the equation $$ x_{1}^{2}x_{2}^{2}x_{3}^{2}x_{4}^{...
- 91
9
votes
1
answer
3k
views
Is every finite Borel measure on a locally compact Hausdorff, $\sigma$-compact and separable space automatically regular?
The conditions stated in the question seem mouthful and a bit arbitrary, so let me provide some backgrounds.
Definition
Let $\mu$ be a Borel measure on a topological space. We say:
$\...
- 83
9
votes
1
answer
1k
views
Luxemburg norm as argument of Young's function: $\Phi\left(\lVert f \rVert_{L^{\Phi}}\right)$
Let $\Phi$ be a Youngs's function, i.e.
$$ \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
9
votes
1
answer
1k
views
Traces of Sobolev spaces
Is there a simple proof of the following fact?
Theorem. Let $\Omega\subset\mathbb{R}^n$ be a bounded and smooth domain. If $n>2$, then $W^{1,n-1}(\partial\Omega)\subset
W^{1-\frac{1}{n},n}(\...
- 28k
9
votes
2
answers
746
views
Characters of irreducible unitary representations of the Poincaré group
Consider Poincare group $\mathrm{ISO}(1,d-1)$, given by $\mathbb R^{1,d-1}\rtimes SO(1,d-1)$ in signature $(1,d-1)$, for some odd $d \geq 3$.
Denote the universal cover of the component connected to ...
- 533
9
votes
2
answers
1k
views
On the definition of "almost-everywhere" for non-complete measure spaces
If $(X,\mathcal{B},\mu)$ is a (non-necessarily complete) measure space, we can give two different notions of a property $P(x)$ that is true almost-everywhere :
(D1) There is a measurable set $A$ ...
- 549
9
votes
3
answers
2k
views
Real analyticity of solution of heat equation
Consider the heat equation $\partial_t u - \Delta u = 0, u(0, x) = u_0$ on a complete (non-compact) Riemannian manifold $M$, may be even $\mathbb{R}^n$. I was wondering, what are some known sufficient ...
- 1,407
9
votes
1
answer
1k
views
Sobolev space for Mixed Dirichlet - Neumann boundary condition
Consider the subset $\Omega\subset \mathbb{R}^N$ with boundary $\partial\Omega$ sufficiently regular and let $\Gamma\subset\partial\Omega$ be a $(N-1)$- dimensional submanifold of $\partial\Omega$. ...
- 253
9
votes
1
answer
338
views
Commuting nets for commuting projections
I think this should not be too difficult, but I am not an expert. I did not get an answer on stackexchange.
Let $A$ be a $C$*-algebra and let $p,q\in A^{**}$ be two commuting projections. Then there ...
- 633
9
votes
2
answers
308
views
Local-to-global inequalities for measures: Brunn-Minkowski, Ahlswede-Daykin, what else?
This question is motivated by an obvious formal analogy between two well-known inequalities:
Log-concavity and Brunn-Minkowski inequality
Let $\mu(dx) := m(x) dx$ be an absolutely continuous ...
- 4,010
9
votes
3
answers
2k
views
2-Wasserstein (optimal transport) and extension to the set of all signed measures
Consider the 2-Wasserstein distance between probability measures $\mu$ and $\nu$ (on $\mathbb{R}^d$), defined as
$$
d_{W_2}(\mu,\nu) = \inf_{\gamma} \Big[\int \|x-y\|^2 d\gamma(x,y)\Big]^{1/2}
$$
...
- 1,731
9
votes
2
answers
1k
views
Hilbert transforms of measures
Given a finite measure $\mu$ on the real line $\mathbb R$, one definition of its Hilbert transform is $(H\mu)(y) =\frac{1}{\pi}(PV)\int \frac{d\mu(x)}{x-y}$ which is known to exist almost everywhere ...
- 91
9
votes
1
answer
2k
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The Invariant Subspace Problem: examples
Question. Is there a concrete example of a bounded linear operator on a Hilbert space for which it is not known if it has a non-trivial closed invariant subspace?
[Added 24.01.2011: According to ...
- 22.3k
9
votes
2
answers
1k
views
Generalization of the positive semidefinite Grothendieck inequality
In a recent paper, S. Khot and A. Naor show a natural generalization of the positive semidefinite Grothendieck's inequality. Grothendieck showed that there exists a constant $K > 0$ such that for ...
- 28.6k
9
votes
1
answer
337
views
Characterizing germs of smooth functions
There's a sheaf of smooth real-valued functions on $\mathbb{R}$, and its germ at $0$ is some vector space $V$. I would like to understand this space. There is a surjective linear map
$$ \phi \colon ...
- 22.3k
9
votes
1
answer
311
views
Continuously varying norms
Let $V$ be a finite-dimensional real vector space with its Euclidean topology. Then all norms on $V$ are equivalent and consequently given two norms $\lVert-\rVert$, $\lVert-\rVert'$, the number
$$
d =...
- 1,153
9
votes
1
answer
221
views
Function spaces satisfying $\mathcal{F}(M\times N)\simeq\mathcal{F}(M)\otimes\mathcal{F}(N)$
Let $M \mapsto\mathcal{F}(M)$ be a map associating topological vector spaces of some type (that I will call "function spaces") to geometric spaces $M$ of some type.
For $M$, I'm mostly ...
- 23.4k
9
votes
1
answer
355
views
Scottish Book Problem 172
The problem is formulated using old terminology and I want to understand what it actually says.
The problem reads: "A space $E$ of type (B) has the property (a) if the weak closure of an ...
- 3,497
9
votes
1
answer
832
views
Baire category theorem for uncountable unions
Any compact Hausdorff space $X$ is a Baire space:
if the set $X$ is a meager set (meaning a countable union of nowhere dense subsets,
also known as a set of first category),
then $X$ is empty.
I am ...
- 37.8k
9
votes
2
answers
777
views
Heat flow, decay of the Fisher information, and $\lambda$-displacement convexity
In the whole post I will work in the flat torus $\mathbb T^d=\mathbb R^d/\mathbb Z^d$ and $\rho$ will stand for any probability measure $\mathcal P(\mathbb T^d)$. This question is strongly related to ...
- 5,380
9
votes
1
answer
596
views
Why is the Berkovich spectrum of a C*-Algebra the same as the Gelfand spectrum?
Let $A = \mathcal{C}(X)$ be a commutative (unital) C*-Algebra. Let $Spec(A)$ denote its Gelfand spectrum
$$ Spec(A) = \{A \rightarrow \mathbb{C} : \text{non-zero *-homomorphism} \} \simeq X. $$
Now ...
- 335
9
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1
answer
460
views
Why should the map $-\Delta^{-1}$ be continuous?
I'm reading an article by Wei-Ming Ni about the existence of solutions for the elliptic problem $$\Delta u +|x|^\lambda |u|^\tau =0,$$
in the unit ball $\Omega$ in dimension $>2$. I'm looking for ...
- 91
9
votes
2
answers
778
views
Rellich's theorem from compact resolvent
On a compact Riemannian manifold, we know that the Laplacian $\Delta$ has compact resolvent. In proving this, one typical way is to use Rellich's theorem about the compact embedding of $H^1(M)$ into $...
- 91
9
votes
1
answer
695
views
What happens to continuous spectrum upon discretization?
Excuse me for a bit of an vague question, but I haven't been able to find a definite answer for this for quite some time. My question is regarding (mostly non-normal )linear operators and their ...
- 91
9
votes
1
answer
2k
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Dual or pre-dual of BV
Was there any relevant work to determine the dual (or more likely the predual) of the space of bounded variation functions $BV(\mathbb{R}^n)$ (I recall the definition : a function in $L^1(\mathbb{R}^n)...
- 1,616
9
votes
1
answer
947
views
On the convergence of the function series $\sum_{n=0}^\infty(-1)^n\frac{f^{(n)}(x)}{n!}x^n$
Let $f$ be a smooth real function defined around origin. If we
differentiate term by term the series
$\hat{f}(x):=\sum_{n=0}^\infty(-1)^n\frac{f^{(n)}(x)}{n!}x^n$, we get $\frac{d}{dx}\hat{f}(x)=0$.
\...
- 107
9
votes
3
answers
1k
views
Why is this operator compact?
Let $D$ be the Dirac-Operator on $\mathbb{R}^n$ or more generally the Dirac spinor bundle $\mathcal{S}\to M$ of a (semi-)Riemannian spin manifold $M$. Then we consider $D$ as an unbouded Operator on $\...
- 229
9
votes
1
answer
784
views
Topological Generalization of Whitney's Extension Theorem
From Wikipedia:
In mathematics, in particular in mathematical analysis, the Whitney extension theorem is a partial converse to Taylor's theorem. Roughly speaking, the theorem asserts that if $A$ is ...
- 8,512
9
votes
1
answer
302
views
What are the points of the algebra of polynomial functions on an arbitrary vector space?
Let $V$ be an arbitrary vector space over some field $\mathbb{K}$ (UPD: of characteristic 0), $V^*=\mathrm{Hom}(V,\mathbb{K})$ its linear dual. Let $\mathrm{Sym}_\mathbb{K}(V^*)$ be the free ...
- 480
9
votes
1
answer
428
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The cardinality of projections of subsets of the Hilbert cube by inner products
I have three related questions.
Question 1: Is there a subset $X$ of the Hilbert cube $[0,1]^{\Bbb N}$ of cardinality continuum, such that for each sequence $a\in [0,1]^{\Bbb N}$ with $\sum a_n$ ...
- 3,104
9
votes
1
answer
219
views
Examples of non-self-induced algebras
Let $A$ be a (possibly non-unital) algebra over $\mathbb C$. We say that $A$ is self-induced if the product map $m:A \otimes_A A \rightarrow A$ is an isomorphism. Here $A \otimes_A A$ is the ...
- 18.7k
9
votes
1
answer
359
views
Relaxation of notion of positive definite function
A function $f:\mathbb{R}\to\mathbb{R}$ is called positive definite (in the semigroup sense) if for all $n\geq 1$ and $x_1,\ldots,x_n\in\mathbb{R}$ pairwise different the matrix $(f(x_i+x_j))_{i,j=1}^n$...
- 3,031
9
votes
1
answer
977
views
Completed tensor product is exact
In the beginning of the 7th chapter of the book "Spectral theory and analytic geometry over non-Archimedean fields" by Vladimir Berkovich one can find the phrase "...tensor product functor is exact on ...
- 332
9
votes
1
answer
544
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Behavior of a Baker-Campbell-Hausdorff problem at infinity
The Baker-Campbell-Hausdorff problem is to obtain $\log(e^{X}e^{Y})$ where $X,Y$ are appropriate operators. The Dynkin series $$\log(e^{tX}e^{tY})=t(X+Y)+\frac{t^2}{2}[X,Y]+o(t^3)$$ gives an expansion ...
- 319
9
votes
2
answers
548
views
What mode of convergence is this?
I'm interested in a new (to me) mode of convergence which is stronger than convergence in measure/probability. I want to know if it has a name and if it is used much in the literature. I will write ...
- 6,287
9
votes
2
answers
553
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Asymptotic behavior of Sturm-Liouville eigenvalues
I have two questions.
Consider the operator $Av = -v'' + a(x)v$ on $I = (0, L)$, with zero Dirichlet condition and $a \in C([0, L])$.
Let $(\lambda_n)$ denote the sequence of eigenvalues of $A$....
- 369
9
votes
1
answer
427
views
A version of von Neumann inequality
Assume that $X,Y,Z$ are three commuting operators acting in a Hilbert space $H$. Assume also that they satisfy following properties:
1) $\|Z\| \le 1$, i.e. $Z$ is a contraction;
2) For any complex ...
- 315
9
votes
3
answers
4k
views
Is there a reference for compact imbedding theory of Hölder space?
This question is posted and unanswered from math.stackexchange.
Suppose $0 < \alpha < \beta$ and $\Omega$ is bounded. Then, the Hölder space $C^\beta(\Omega)$ is compactly imbedded to $C^\alpha(...
- 1,399
9
votes
1
answer
544
views
Question on Hilbert Manifolds
I have a very basic question on Hilbert manifolds.
Consider the Hilbert space
$$
\mathcal{H}:= L^2(S^1)
$$
with $S^1$ the unit circle.
On $\mathcal{H}$ let us introduce the equivalence relation
$$
...
- 233
9
votes
1
answer
450
views
A question on infinite dimensional Gaussian measure and affine tranformations.
Let $\gamma_\infty$ denote the product Gaussian measure on $\mathbb{R}^\mathbb{N}$. Which $a,b \geq 0$ satisfy that for every Borel set $K\subseteq \mathbb{R}^\mathbb{N}$ of positive measure, $a K + ...
- 3,547
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 ...
- 949
9
votes
1
answer
384
views
Comparing two $\sigma$-algebras on $B(\ell^1)$
Let us consider $B(\ell^1)$, bounded linear operators on $\ell^1$. We recall the weak operator topology, denoted by $w$, on $B(\ell^1)$ is determined as follow
$$w-\lim T_i=T \Longleftrightarrow \...
- 4,058
9
votes
1
answer
415
views
Relationship between Harish-Chandra Schwartz space and more generic Schwartz spaces
If $G$ is a connected semisimple Lie group with finite center, Harish-Chandra defined a Schwartz space of rapidly decreasing functions on $G$ as the space of $\mathrm{C}^\infty$ functions defined by ...
- 3,816
9
votes
1
answer
570
views
Elements of $L^p$ and nice representatives of equivalence classes
Considering $L^p$ $( 1 \leq p < \infty)$ as a normed vector space, each element of $L^p$ is actually an Equivalent class. Take $[f] \in L^p $ as an Equivalent class, What is the Nicest possible ...
- 369