All Questions
10,240 questions
7
votes
0
answers
300
views
Generalized Skorokhod spaces
Skorokhod spaces of càdlàg functions are an extremely useful setting to describe stochastic processes. I'd like to understand the Skorokhod topology from a pure topological point of view, without ...
- 8,512
7
votes
0
answers
355
views
An $L^{\infty}$ version of principal component analysis?
I have a $k$ by $n$ matrix $A$, with $k \ll n$. In case it helps, the $k$ rows are orthonormal.
I'm interested in finding a $k$ by $k$ orthogonal matrix $M$ so as to maximize the $L^{\infty}$ norms ...
- 193
7
votes
1
answer
331
views
States/functionals on crossed product C*-algebras
Let $A$ be a C*-algebra, $\alpha$ a strongly continuous automorphic action by a locally compact group $G$ on $A$, and consider the crossed product $A\rtimes_\alpha G$. I am looking for references ...
- 1,017
7
votes
0
answers
266
views
Problem with Shelah and Stern's paper on the Hanf number of the theory of Banach spaces
I have been trying to understand "The Hanf number of the first order theory of Banach spaces" by Shelah and Stern (Trans. AMS 244 (1978) 147-241). They construct a normed space $M$ from a Hilbert ...
- 879
7
votes
0
answers
624
views
"Liftings" of L^\infty functions
This is motivated by this question: Is there an inclusion of $L_\infty(G)$ into $C_0(G)^{**}$? and Bill Johnson's comments there.
Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon ...
- 18.7k
7
votes
0
answers
140
views
integral transforms defined by polygons
Are there any literature on operators of the form
\[ (Tf)(x) = \int K(x,y)f(y), dy\]
where $K(x,y)$ is the characteristic function of a polygon in $\mathbb{R}^2$. I would like to know if the spectrum ...
- 22.8k
7
votes
0
answers
161
views
Seeking reference - criterion for the existence of a positive linear functional on an ordered vector space below a given function
The following surely appears somewhere, I would greatly appreciate a reference. (The aim is to get a measure via Riesz representation, but that has nothing to do with the statement.)
Let $X$ be an ...
- 577
7
votes
2
answers
1k
views
Weighted Poincaré inequality
Consider a probability distribution $\pi$ with density $e^{-H(x)}$ on $\mathbb{R}$. Let us say that there is a Poincaré inequality with weight $w$ if for any smooth function $\phi$ satisfying $\int \...
- 2,133
7
votes
0
answers
1k
views
Reference request: Arzela-Ascoli theorem for smooth Hölder norms
Could anyone suggest a textbook account of the Arzela-Ascoli theorem for $C^{k,\alpha}$ norms?
- 29.1k
7
votes
0
answers
4k
views
Explicit element of $(\ell^{\infty})^* - \ell^1$? [duplicate]
Possible Duplicate:
What’s an example of a space that needs the Hahn-Banach Theorem?
It is well known that the dual of $\ell^{\infty}$ properly contains $\ell^1$ (over $\mathbb{N}$, say). ...
- 25.6k
6
votes
3
answers
2k
views
reflexive banach space
I want to ask this non-expert question:
What does it mean geometrically for a Banach space to be reflexive?
Well, we could say a Banach space is reflexive iff unit ball is weakly compact. Or some ...
- 351
6
votes
3
answers
3k
views
Why isn't the theorem of approximation applicable in Banach spaces?
Let X be a Hilbert space, A a convex, closed subset of X. Then there exists for every x in X exactly one best approximation in A, that is there exists a y in A such that || x-y || = d(x,A) = inf { || ...
- 61
6
votes
1
answer
938
views
Convergence of Fourier series
Say $f \in L^p[a,b]$, with $p \in \mathbb{N}, p > 1 $. Does its Fourier Series converge in the metric space $L^p[a,b]$? Does the series converge pointwise? And at which conditions?
Say now $p = 1$, ...
- 117
6
votes
3
answers
415
views
Examples of amenable Banach algebras which have non-amenable subalgebra
I am looking for examples of amenable Banach algebras which have non-amenable subalgebra
I know
1: Each amenable Banach algebra has a bounded approximate identity
2: If $I$ be a closed ideal in an ...
- 823
6
votes
3
answers
1k
views
Orthonormal basis in $W^{1,2}([0,1])$
Consider the Hilbertspace $W^{1,2}([0,1])$ (i.e. Sobolev space) with the standard inner product which is defined by: $(f,g) = (f,g)_{L^{2}([0,1])} + (f',g')_{L^{2}([0,1])}$. Here $[0,1]$ is not ...
- 63
6
votes
3
answers
1k
views
Topological vector spaces that are isomorphic to their duals
After reviewing the (locally convex)
topological vector spaces that I know,
the only examples I could find where there is an isomorphism from the
space to its (anti)dual, are Hilbert spaces.
So my ...
- 357
6
votes
2
answers
1k
views
Exercise 8.13 - Brezis
Let $1 \leq p < \infty$ and $u \in W^{1,p}(\mathbb{R}$). Set
$$
D_{h}u(x) = \frac{1}{h}(u(x+h) - u(x)), \ \ x \in \mathbb{R}, h> 0
$$
Show that $D_{h}u \to u'$ in $L^{p}(\mathbb{R}$) as $h \to ...
- 241
6
votes
3
answers
452
views
Norm estimation of identity plus two non-commuting self-adjoint operators
This is a problem that I have been stuck for a few months.
Let $X$ be a Hilbert space and $A:B:X\to X$ be two non-commuting semi-positive self-adjoint bounded linear operators. Is it true that
$$\|(I+...
- 503
6
votes
2
answers
4k
views
Question about Schauder bases in C([0,1]).
I would like to check a statement about Schauder bases in $C([0,1])$ to be sure that I don't lie to my students on Monday. The statement(s) that I would like to check are:
The family of monomials $\{...
- 26.8k
6
votes
4
answers
1k
views
Convex set with no interior contained in hyperplane?
Let $K$ be a convex set in a normed space $X$. Assume that $int(K)=\emptyset$ (norm topology). Must $K$ be contained in some (affine) hyperplane?
It's fairly easy to see that this is true in $ℝ^n$, ...
- 165
6
votes
1
answer
1k
views
Compactness in Sobolev spaces
I was wondering whether the set $\lbrace f\in H_0^1(\Omega)|\|f\|_{L^\infty(\Omega)}\leq 1\rbrace$ is compact in $H_0^1(\Omega)$ or not. Here $\Omega$ is a convex domain in $\mathbb{R}^3$ with ...
- 63
6
votes
3
answers
852
views
Are nuclear operators closed under extensions?
Given $X_i, Y_i$ Banach spaces, $f_j, g_j, T_i$ bounded linear operators for $i=1,2,3$ and $j=1,2$. We have the following diagram
$\require{AMScd}$
\begin{CD}
0 @>>> X_1 @>f_1>> X_2 ...
- 1,338
6
votes
3
answers
427
views
Point-wise limit of finite valued functions
Let $X$ be a second countable topological vector space. Does there exist any sequence of finite valued functions $f_n\colon X\to X$ converging point-wise to the identity mapping on $X$?
- 4,058
6
votes
3
answers
2k
views
Sequential topological vector spaces
Since I'm dealing with the distinction between sequential continuous and continuous maps at the moment I came to ask myself once again what can be said about spaces where these two notions agree (...
- 9,650
6
votes
5
answers
1k
views
smooth Gelfand-duality
Assume $M$ is a compact smooth manifold (without boundary). What can we say about the spectrum of the $\mathbb{R}$-algebra $A=C^{\infty}(M)$? The elements of $M$ give rise to rational points of $A$, ...
- 63.1k
6
votes
2
answers
1k
views
Vector-Valued Stone-Weierstrass Theorem?
The standard statement of the Stone-Weierstrass theorem is:
Let $X$ be compact Hausdorff topological space, and $\mathcal{A}$ a subalgebra of the continuous functions from $X$ to $\mathbb{R}$ which ...
- 263
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 ...
- 11.3k
6
votes
4
answers
7k
views
Why do we want to have orthogonal bases in decompositions?
In the decompositions I encountered so far, we all had orthogonal set of bases. For example in Singular Value Decomposition, we had orthogonal singular right and left vectors, in [discrete] cosine ...
- 497
6
votes
2
answers
572
views
Why are $\Gamma_0$ functions called this
It is very common to indicate with $\Gamma_0(A)$ the set of lower semicontinuous convex functions from $A$ to $(-\infty,+\infty]$ with nonempty domain. An example of usage of this notation can be ...
- 71
6
votes
2
answers
690
views
Can $C^*$-algebra of continuous functions on $R^n$ ($S^n$) be characterized alternatively?
Dictionary between algebra and geometry is somewhat one of the main concepts in modern mathematics. So commutative $C^*$ algebras are one-to-one with locally compact Hausdorff spaces.
So it is ...
- 24.9k
6
votes
2
answers
3k
views
Fourier series representing a continuous function?
This is maybe not really research level, but I have not found anything in the literature, and asking on math.stackexchange wasn't successful either.
Fourier series define an isometry $L^2(\mathbb{Z}) ...
- 976
6
votes
3
answers
993
views
Invertible operator with countable spectrum
Let $H$ be a separable Hilbert space and $A$ is an invertible bounded operator on $H$. Can we approximate $A$ with an invertible operator $B$ such that $sp(B)$ is a countable set?
Motivation:
...
- 366
6
votes
3
answers
3k
views
Can the adjoint of the exterior derivative in semi-Riemannian geometry be defined without the Hodge * operator?
The adjoint of the exterior derivarive is defined by
$\delta:=(-1)^k\ast^{-1}d\ast$,
but I need a way which avoids the Hodge $\ast$ operator.
Is there another definition?
For example, for ...
- 4,312
6
votes
3
answers
964
views
Convolution of $L^2$ functions
Let $u\in L^2(\mathbb R^n)$: then $u\ast u$ is a bounded continuous function. Let me assume now that $u\ast u$ is compactly supported. Is there anything relevant that could be said on the support of $...
- 16.2k
6
votes
3
answers
508
views
Proof of the Schauder Lemma
Schauder's Lemma in functional analysis states the following:
Let $E$ and $F$ be metrizable locally convex topological vector spaces, and let $E$ be Fréchet. Then if the linear continuous map $A:E\...
- 1,247
6
votes
7
answers
8k
views
Existence of an extreme point of a compact convex set
The Krein-Milman theorem shows that a compact convex set in a Hausdorff locally convex topological vector space is the convex hull of its extreme points.
It seems this implies that a compact convex ...
- 169
6
votes
3
answers
3k
views
Non-empty resolvent set, then operator closed?
On Hilbert spaces, the following is true:
Let $T$ be a densely-defined linear operator with non-empty resolvent set, then $T$ is closed.
The obvious proof I see to show this uses explicitly the ...
- 115
6
votes
2
answers
1k
views
Vector Fields in a Riemannian Manifold
Suppose $(M,g)$ is a Riemannian manifold.
Is there a way to classify manifolds where there exists a vector field that commutes with the laplace beltrami operator?
Thanks
- 4,115
6
votes
4
answers
1k
views
Recovering a measure from its moments
Suppose we are given moments of a measure on the interval [0,1]. Is there some practical way to recover the measure itself? I am particularly interested in the case where the measure density is given ...
6
votes
1
answer
1k
views
Is there any nontrivial characterization of weakly differentiable functions?
When $f\in L_\text{loc}^1$, it's distributional derivative can be defined as $D_{f'}\in\mathfrak{D}'$, such that $D_{f'}(\varphi)=-\int f\varphi'$ for all $\varphi\in\mathfrak{D}$, where $\mathfrak{D}$...
6
votes
2
answers
881
views
Summation of bounded sequences
Let $b_{n,j}\in \mathbb{C}$ for each $n,j\in \mathbb{N}$. I was wondering if there is some characterization of those $b_{n,j}$ such that for all bounded sequences $s_j\in \mathbb{C}, j\in \mathbb{N}$, ...
- 163
6
votes
2
answers
749
views
Transpose of unbounded operators between Banach spaces.
Let $X$ and $Y$ be Banach spaces, and let $L : X \rightarrow Y$ be a unbounded operator with dense domain $\operatorname{dom}(L)$. We can then talk about the transposed operator
$L' : \operatorname{...
- 5,327
6
votes
2
answers
1k
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An element of $(L^{\infty})^*$ which does not seem to be a finitely additive abs. cont. measure.
Hi everyone,
I have a question which I am quite stumped on. Consider the linear functional $l(f) = f(0)$ defined on $C([-1,1])$. By Hahn-Banach this linear functional can be extended to one on all ...
- 2,641
6
votes
3
answers
1k
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How can I embed an N-points metric space to a hypercube with low distortion?
I have a N-point metric space defined by the pairwise distance matrix. I want to encode these N points with binary strings, i.e. each point will be mapped to a vertex in a hypercube. The lengths of ...
- 171
6
votes
2
answers
1k
views
Closed, complemented subspaces of $l^1(X)$ when $X$ is uncountable
... are all isomorphic to $l^1$ on some other index set. At least, that much I "know" from 2nd-hand sources, since the original proof is apparently in a paper of Köthe from the 1930s 1960s (in ...
- 25.8k
6
votes
2
answers
325
views
Nonvanishing section of infinite-dimensional tautological bundle
Let $H$ be a real or complex Hilbert space. In the case where $H$ is infinite-dimensional, let us define a half-dimensional subspace as a subspace $W \subset H$ such that both $W$ and $W^\perp$ have ...
- 9,853
6
votes
2
answers
1k
views
The contractivity of the heat semigroup in $L^p$ spaces
Let $M$ be a Riemannian manifold. By functional calculus, it is immediate to show that the heat semigroup is a contraction in $L^2(M)$. I can also show that it is a contraction in any $L^p(M)$ with $p ...
- 5,407
6
votes
2
answers
5k
views
Domain of definition of Laplace Operator on $L^2$
I'm trying to combine two ways of looking at the Laplacian $\Delta$ on $\mathbb R^n$ (and on other domains).
Firstly, it is well known that this operator is essentially self-adjoint on $C_c^\infty(\...
6
votes
6
answers
2k
views
Application of bounded spectral theory.
I'm trying to gain some intuition for the usefullness of the spectral theory for bounded self adjoint operators. I work in PDE and any interesting applications/examples I've ever encountered are ...
- 2,641
6
votes
1
answer
403
views
Do acyclic amenable groups exist?
Is there an example of a nontrivial discrete amenable group with vanishing integral homology?
To put the question in contrapositive. Given arbitrary acyclic group $Q$, is there some reason for the ...
- 4,600