All Questions
4,446 questions with no upvoted or accepted answers
3
votes
0
answers
84
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Application and relevance of Sobolev gradients
The Sobolev gradient concept has been developed in the 1970s, with a first publication in 1985, and an introduction can be found at: Ranka
I would like to learn how strong the impact of Sobolev ...
- 5,327
3
votes
0
answers
301
views
What information about a locally compact group $G$ is encoded in $C_r^\ast(G)$ which is not in $L^1(G)$?
Let $G$ be a locally compact group and let $ C_r^\ast(G) $ denote its reduced group $C^\ast$-algebra. Many features of a $G$ can be realized from $L^1(G)$ or $C_r^\ast(G)$. For example, $G$ is ...
user23860
3
votes
0
answers
176
views
Extending a Hilbert space isometrically
Let $H$ be a Hilbert space, and let $X$ be a topological vector space.
Under what conditions on the topologies of $X$ and $H$ does there exist an injective, continuous linear map $f : H \to X$?
...
- 8,512
3
votes
0
answers
381
views
Extension divergence-free, curl-converging vector field
Hi.
Consider a smooth open Set $\Omega\subset\mathbb{R}^3$ and a bounded sequence of vector fields $(u_n)_n \in L^2(\Omega)$ having $0$ divergence. I know how to extend this sequence to the whole ...
- 3,425
3
votes
0
answers
145
views
Growth of inner functions on the disk
Recall that an inner function on the disk $D$ is a bounded analytic function on $D$ having radial limits of modulus one almost everywhere.
There has been many works on the growth of the inner ...
- 769
3
votes
0
answers
183
views
Is the construction of ring C*-algebra functorial?
Cunz and Li defined defined C*-algebras for arbitrary rings with of course some condition. One can look at their article (http://arxiv.org/abs/0905.4861). My question: is the construction functorial? ...
- 95
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 ...
3
votes
0
answers
185
views
spectrum of a polygon and zeta function
Let $\Delta(x,y) = 1,0$ according to whether $(x,y)$ is in some polygon (symmetric with respect to the diagonal axis).
E.g. The convex hull of three points (taken from a paper on dominoes)
$$ \...
- 22.8k
3
votes
0
answers
867
views
The inductive and projective limits of compact Hausdorff topological groups
Are there conditions known under which the inductive or projective limit of a family of compact Hausdorff topological groups is compact? (For instance, such a result is valid for the projective limit ...
- 5,407
3
votes
0
answers
163
views
Isometric automorphism of $c_0$ different than coordinate permutation
Does there exist an isometric automorphism of $c_0$ which is not a permutation of coordinates?
- 311
3
votes
0
answers
269
views
Continuous selection given both upper and lower hemicontinuity
Suppose that $X,Y$ are compact metric spaces, and $g:X\rightarrow Y$ is a multivalued mapping that is both upper and lower hemicontinuous. Is there a single valued continuous selection of $g$? If it ...
- 203
3
votes
0
answers
209
views
A maximum of a function
When studying the $\text{UMD}$ constants of spaces like $L_{p_1}(L_{p_2}(\cdots (L_{p_n})\cdots))$, I encounter the following question: Let $\alpha > 0$, define $$C(\alpha) : = \sup_{a > 0, b>...
- 769
3
votes
0
answers
396
views
Norm estimate for Moore-Penrose pseudo-inverse of $i^\ast T i$
Let $G$ and $H$ be Hilbert spaces, let $i : G \rightarrow H$ be an isometric inclusion (so $G$ is a subspace of $H$) and let $T : H \rightarrow H$ be a bounded linear operator with closed range.
That ...
- 5,327
3
votes
0
answers
302
views
Dense subalgebras of von Neumann algebras and increasing nets
[Question previously asked on Math.SE]
Let $N$ be a von Neumann algebra, and $A$ be a dense $∗$-subalgebra of $N$ (in the ultraweak topology) with $A''=N$. Is it true that:
For any $x∈N^+$, there ...
- 33
3
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0
answers
254
views
Ways to establish equality of measures on locally compact spaces
Let $M$ be a locally compact space and $\mu$ be some probability measure on $M$. Let $y^\ast \in M$, $f(x,y)$ be a real continuous bounded function $M \times M \to \mathbb{R}$. Consider an equality
$$
...
- 1,329
3
votes
0
answers
409
views
Continuous function sort
If you have a real-valued function f(x), positive, continuous and bounded on some interval, then what kind of transform would convert this to a monotonic function g(x) on that interval analogously to ...
- 529
3
votes
0
answers
217
views
Is this integral operator about Stokes' Flow compact?
Consider the following integral operator $\mathcal{A}$ on [EDITED: continuous vector function $f=(f_i):\partial S\to{\mathbb R}^3$]:
$$
({\mathcal A}f)_j(x_0):=\int_{\partial S}\sum_{i=1}^3 f_i(x)G_{...
user14319
3
votes
0
answers
188
views
Does the existence of an asymtpotic density imply the existence of a measure on infinite dimensional (path) space?
This question is related to the following question
Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)?
A couple of authors have observed that composing a ...
- 265
3
votes
0
answers
456
views
Morphism of von Neumann Algebras
Hello,
Is there a counterexample to the following statement:
let $A,B$ two von Neumann algebras, every morphism $A \rightarrow B$ of $C^* $-algebras is a $W^*$-homomorphism ?
( a $W^* $-...
- 663
3
votes
0
answers
877
views
The "pullback presheaf" and the proper base change theorem in topology
Let $f:X\rightarrow Y$ be a continuous map of topological spaces and let $\mathcal{F}$
be a sheaf (say of abelian groups to fix the idea) on $Y$. Define the following rule on open sets of $X$:
$$
V\...
- 7,609
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
3
votes
0
answers
277
views
For METRIZABLE spaces, do the Banach classes and Baire classes coincide?
In this paper: 'Borel structures for Function spaces' by Robert Aumann,
http://projecteuclid.org/euclid.ijm/1255631584
Aumann claims that when X and Y are metric spaces (among other things), the ...
- 456
3
votes
1
answer
1k
views
Borel-Cantelli lemma for general measure spaces (those with infinite measure)
The Borel-Cantelli lemma is often stated for a probability space or spaces with finite measure.
But it seems to me that it still holds if the space $X$ is of infinite measure. I seem to be able to ...
- 31
3
votes
0
answers
385
views
Off-diagonal asymptotic expansion of the Bergman kernel on hyperbolic Riemann surfaces
Let $X$ be a compact Riemann surface of genus at least 2. Let $K$ denote the canonical line bundle, and $E$ be any vector bundle. Let $P^{(m)}$ be the projection map from the space $L^2(X,K^mE)$ of ...
user2529
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
3
votes
0
answers
251
views
What is the origin of the metrization problem for compact convex sets?
The following is an ``old question in analysis:''
Is it true that every perfectly normal compact convex subset of a locally convex topological vector space is metrizable?
Here perfectly normal means ...
- 3,547
3
votes
0
answers
356
views
Stability of convex sets w.r.t. integration over [0,1]
In the preprint on pages 19−20, first using Hahn−Banach, one proves
Lemma 38. For any closed convex set $U$ in a real Hausdorff locally convex space $E$ and for any Riemann integrable $\gamma:[0,1]\...
- 3,584
3
votes
0
answers
223
views
Extension of positive operators and Bauer-Namioka
When $X$ is a vector subspace of an ordered vector space $A$, any positive linear functional $f: X \to R$ extends to all of $A$ as a positive linear functional provided one can find a nonvoid, ...
- 31
3
votes
0
answers
1k
views
weak regularity conditions for regions to assure boundary of measure zero
Let $\Omega \subset \mathbb{R}^d$ be a region ( bounded, simply connected, open set ). What are some regularity conditions to assure the boundary $\partial\Omega$ is a set of (lebesgue-)measure zero? ...
- 131
3
votes
0
answers
168
views
Mapping into Hurewicz cofibrations.
In Strøm's paper "The Homotopy Category is a Homotopy Category" he proves
(Lemma 4) that if $Y$ is compact and if $i:A\to X$ is a cofibration, then the induced map
$$
i_*: A^Y \to X^Y
$$
is also a ...
- 12.5k
3
votes
0
answers
130
views
Positive block matrices over tensor algebras
Let $A$ be a unital C*-algebra. A positive block matrix in $M_2(A)$ must have the form
$$ \begin{pmatrix} a & a^{1/2} x b^{1/2} \\ b^{1/2} x^* a^{1/2} & b \end{pmatrix}, $$
where $a,b$ are ...
- 18.7k
3
votes
0
answers
294
views
Monomorphisms in geometry
What is known about monomorphisms in the following categories:
Schemes
Complex manifolds
$C^\infty$--manifolds
and any other kinds of geometric objects that you might think of.
How do we choose a ...
- 123
3
votes
0
answers
126
views
More on continuous images of dense orders
In this question I asked if there was an analogue of connectedness which applied to dense orders which were not required to be complete. Between them, Noah and Joel showed that every (infinite) ...
- 3,612
3
votes
0
answers
267
views
Maps of loop spaces with infinity-bounded differential.
I am currently working with loop spaces of manifold and finite dimensional manifolds approximating these and the following comes up very naturally:
In the following piece-wise smooth means smooth on ...
- 2,590
3
votes
0
answers
721
views
What is the horn torus homeomorphic to?
Is the horn torus homeomorphic to some other well known object? In particular, the standard torus can be described by a square with collapsed edges. What about the horn torus?
- 1,638
3
votes
1
answer
355
views
convex function with distributional Hessian $D^2 f \leqslant \lambda$, $\lambda$-concave?
Let $f:R^n \to R$ be convex (may not $C^1$), $$[D^2f]=[D^2f]_{ac}+[D^2f]_s=[h_{ij}] L^n+[D^2f]_s$$ is the Lebesgue decomposition of the Hessian matrix. Where $[h_{ij}]$ is the density w.r.t the ...
- 471
3
votes
1
answer
310
views
Do cyclic product vectors generatating irreducible representation of a Lie group come from a unique orbit?
Consider a Hilbert space $\mathcal{H}$ which is a carrier space of a unitary, irreducible and strongly continuous representation $\Pi$ of a Lie group $G$. Let $\Pi\otimes \Pi$ denote the corresponding ...
- 965
3
votes
1
answer
342
views
When does $\lVert T_n \rVert_p \to \lVert T_0 \rVert_p$ imply $\lVert T_n - T_0 \rVert_p \to 0$?
maybe this question is too elementary but I could not find any results in the functional analysis textbooks I own:
For separable Hilbert spaces $H$, $K$, let $S_p(H,K)$ be the $p$th Schatten class of ...
- 617
3
votes
1
answer
441
views
In which topological spaces does the existence of a loop not contractible to a point imply there is a non-contractible simple loop also?
In another MathOverflow post I asked: In a topological space if there exists a loop that cannot be contracted to a point does there exist a simple loop that cannot be contracted also?
Note that ...
- 4,862
3
votes
1
answer
791
views
Real part of eigenvalues and Laplacian
I am working on imaging and I am a bit puzzled by the behaviour of this matrix:
$$A:=\left(
\begin{array}{cccccc}
1 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 &...
user136033
2
votes
0
answers
86
views
Is there a natural topology for subsets of a fixed topological space?
This question is an extension/clarification of the question: Is there a natural topology for sets of topological spaces?
The Hausdorff distance assigns a distance to any two subspaces $X, Y$ of a ...
- 719
2
votes
0
answers
120
views
Uncertainty principle: minimize $\int_{-\infty}^\infty |t| |\widehat{f}(t)|^2 dt$ for $f$ of compact support
This is a question of uncertainty-principle type stemming from Eigenvalue of a convolution and a restriction?
Let $f:\mathbb{R}\to \mathbb{R}$ be even, absolutely continuous and supported in $[-\frac{...
- 20.2k
2
votes
0
answers
81
views
Function that is (essentially) a self-convolution but not a multiple of a self-convolution
Call a function $F:\mathbb{R}\to C$ nice if it is of the form $F = f\ast \tilde{f}$, where $\tilde{f}(x) = \overline{f(-x)}$. (Of course nice functions are precisely those whose Fourier transform is ...
- 20.2k
2
votes
0
answers
230
views
A deceptively simple regularity problem for functions on the plane
By various meanderings and toying with simpler problems, my current research has lead me to the following quite straightforward question, which I am wholly unable to answer:
Consider a twice ...
- 989
2
votes
0
answers
68
views
Bessel spaces and Triebel Lizorkin
It is known that bessel potential spaces $H^{s,p}$ coincide with Triebel-Lizorkin spaces $F^{s}_{p,2}$ for $s\in \mathbb{R}$ and $1<p<\infty$. Im wondering what can be said por $p=1$ and $p=\...
2
votes
0
answers
159
views
About the 7.3.5. Corollary of the book "Measure Theory" by V.I. Bogachev
According to the 7.3.5. Corollary of the book "Measure Theory" by V.I. Bogachev we have the following result:
Let $(X,\tau)$ be a completely regular space and let $\Gamma$ be a
family of ...
- 649
2
votes
0
answers
194
views
Functions such that the *integral* of the Fourier transform is non-negative?
Let $f:\mathbb{R}\to \mathbb{R}$ be in $L^1$, with its Fourier transform $\widehat{f}$ also in $L^1$. What is a necessary and sufficient condition on $f$ so that
$$\int_{-\infty}^x \widehat{f}(t) dt \...
- 20.2k
2
votes
0
answers
52
views
On distributions and kernels
Let $U\subset\mathbb{R}^{d}$ be an open set and consider $X=\mathbb{R}\times U$. Now, lets consider a smooth (regular) kernel $k_{A}\in C^{\infty}(X\times X)$ and corresponding continuous operator $A:...
- 1,429
2
votes
0
answers
70
views
Is the hypothesis "uniformly equivalent" needed?
I am reading S. Shimorin's paper titled Complete Nevanlinna-Pick property of Dirichlet-type spaces. My question concerns Lemma 2.3. which is as follows:
Assume $\mathscr{H}$ is a Hilbert space of ...
- 151
2
votes
0
answers
43
views
Distributions and time-kernels
Let $U\subset\mathbb{R}^{d}$ be an open subset and set $M:=I\times U$, where $I=(a,b)\subset\mathbb{R}$ is some open subset. Lets consider a linear operator $B:C^{\infty}_{c}(M)\to C^{\infty}(M)$ that ...
- 1,429