All Questions
10,050 questions
3
votes
2
answers
904
views
Corona Theorem in several variables
Hallo,
I have read about the Corona Theorem (see link:http://en.wikipedia.org/wiki/Corona_theorem). From this one ca deduce that: Let $f_{1}, ..., f_{n}$ be holomorphic bounded functions on the unit ...
- 339
0
votes
2
answers
765
views
About a generalization of the Radon Nikodym Theorem
Im trying to prove a generalization of the Radon Nykodym theorem, but im having troubles even for finite measures, could someone help?
Let $\mu$ and $\nu$ two $\sigma$-finite measures in $\(X,\...
- 11
6
votes
4
answers
1k
views
Reference for integral of functions taking values in a topological vector space.
(Note that I am interested in the Gelfand-Pettis integral specifically, as opposed to, for example, the Bochner integral.) I have tried Googling things like "integral topological vector space", "...
- 1,270
4
votes
1
answer
598
views
What sort of manifold is PU(H)?
The space underlying the projective unitary group of a separable, infinite-dimensional Hilbert space has a number of topologies, so for the purposes of this question, pick you favourite and answer for ...
- 35.5k
3
votes
1
answer
445
views
Bohr sets, Coin-flip sets and Roth's theorem
I have been learning about Roth's theorem, trying to understand how Fourier series and dynamical systems (or even graph theory and binary sequences)are involved in counting arithmetic sequences in ...
- 22.8k
8
votes
3
answers
1k
views
Relating a Polynomial equation to the characteristic equation of a Hermitian matrix
This question arose out of mere curiosity. Given a polynomial equation and I happen to know that its roots are real (but not the roots itself). Does it mean it is the characteristic equation of a ...
- 1,421
7
votes
3
answers
6k
views
Integral kernel for the resolvent of the laplace operator
Consider the Laplace operator defined in the biggest possible subset of $L^2(\mathbb{R}^2)$ and let $z \in \mathbb{C}\backslash\mathbb{R}$. Therefore $z \notin \sigma (\Delta)$ the spectrum of $\Delta$...
- 225
4
votes
0
answers
158
views
Does this construction yield an injective hull ?
Let $K$ be an object of $\mathbf{CHaus}$, the category of compact Hausdorff spaces, and $K \xrightarrow{\ \ \sigma \ \ } K$ be an involutory morphism without fixed points. Define $C^{\sigma}(K)$ as ...
- 7,249
4
votes
4
answers
1k
views
Continuous pointwise ergodic theorem?
Let $\Phi$ be a homeomorphism of a compact metric space $M$
which preserves a regular Borel
probability measure $\mu$.(`Regular' $\mu(U) > 0$, if U open. )
Under these hypothesis, I have two ...
- 8,336
2
votes
1
answer
637
views
Topological properties of SpecMax(A)
We consider $A = C_{b}(X)$, the ring of continuous bounded functions on a completely regular space $X$. Let $\DeclareMathOperator{\SpecMax}{SpecMax} \SpecMax(A)$ be the set of maximal ideals of $A$ ...
- 933
4
votes
1
answer
419
views
Pitt's theorem for non-separable $\ell_p$ spaces
A short variant of Pitt's theorem is the followig: for $1\leq p < r <\infty$ holds
$$
\mathcal{B}(\ell_r(\mathbb{N}),\ell_p(\mathbb{N}))=\mathcal{K}(\ell_r(\mathbb{N}),\ell_p(\mathbb{N}))
$$
Now ...
- 1,697
6
votes
1
answer
428
views
Poincaré lemma in infinite dimensions
Hi everyone,
Is the Poincaré lemma true in infinite dimensions?
Here's a precise statement:
Let $X$ be a Banach (or maybe Hilbert) vector space, $U$ a simply connected open set in $X$. Is it true ...
- 1,347
0
votes
0
answers
80
views
relationship between different function classes
I was wondering if there is a survey of relationship between several different well-studied function classes ?
ps - The question may be vague but I am looking for something along the lines of - http:/...
- 1
8
votes
1
answer
292
views
A definition of non-commutative metrisable space
If $X$ is a compact metrisable space, a metric $d$ on $X$ can be take as an element of $C(X\times X)$ such that
(1) $ev_x\otimes ev_y (d)=d(x,y)\geq 0$ for all $x,y\in X$ (Non-negativity).
(2) $...
- 831
0
votes
0
answers
218
views
Series of linear maps: on a paper by Evans and Hanche-Olsen
I was reading this paper by Evans and Hanche-Olsen. In theorem 2, there are six equivalent statements given. I write just two of them, which I want to use.
Let $L$ be a bounded self-adjoint
...
- 421
9
votes
1
answer
280
views
Fredholm theory on Fr\'echet spaces
Dear everybody,
In my study of the classial Fredholm theory on Banach spaces, I am interested in the corresponding Fredholm theory on Fr\'echet spaces. But it seems to me that there is
little ...
- 515
12
votes
2
answers
3k
views
Does there exist an isometry between $L^p$ and $l^p$?
The motivation is simple, as it is trivially right when $p=2$. When considering the duality between $L^p$ ($l^p$) and $L^q$ ($l^q$) when $p$ and $q$ are conjugate in the sense that $1/p+1/q=1$, I ...
- 619
8
votes
3
answers
485
views
Does the metric space of compact metric spaces satisfy the binary intersection property?
A metric space $Y$ has the binary intersection property provided that whenever a collection of closed balls in $Y$ intersects pairwise, then there is a common intersection point.
Does the metric ...
- 15.5k
0
votes
1
answer
1k
views
Showing a coercivity condition for this bilinear form
Suppose $\Omega \subset \mathbb{R}^n$ is a compact domain. Let $f$ and $J$ (and also $\frac 1J$) be $C^1$ functions on $\Omega$. Consider the bilinear form $a:H^1(\Omega) \times H^1(\Omega) \to \...
- 107
0
votes
1
answer
383
views
injection with sobolev space
Let $\Omega $ be a bounded open subset of $R^n,\; n\ge 1.$
I m asking about the existence of a subregion $\omega\subset \Omega$ such that the map $y\to y|_\omega $ from $H^2(\Omega)$ into $L^\infty(\...
- 51
6
votes
2
answers
426
views
Ultrapowers of operators
Can we prove that for each infinite dimensional Banach space $X$ and any free ultrafilter (possibly over uncountable set of indices) $\mathcal{U}$ the obvious embedding
$$({\mathcal{L}(X)})_{\mathcal{...
- 143
3
votes
1
answer
3k
views
Choice of Lipschitz constant for proximal gradient optimization
I'm trying to use proximal gradient methods (Forward-Backwards Splitting and FISTA) to minimize a function
$f(B) = \frac{1}{2}|| XB - Y||_F^2 + \frac{\gamma}{2}||B C^T||_F^2$, where $X \in \mathbb{R}...
- 205
0
votes
5
answers
1k
views
the intersection of a sequence of measurable sets
If $\Omega$ is a bounded open set in $R^n$, $\Omega_j\subset\Omega$, and $|\Omega_j|\geq\epsilon$, which $\epsilon$ is a constant. Can we say there is a subsequence $\Omega_{j_k}$ of $\Omega_j$, such ...
- 25
0
votes
2
answers
205
views
ANR Subsets of banach spaces
I need a reference for conditions on a closed subspace of a Banach space to have the homotopy type of an ANR.
5
votes
0
answers
157
views
Containment of an element to an operator system
This question will probably appeal to people in operator systems theory as it is very much related. However, I'm interested in down-to-earth concrete systems with finite dimensional Hilbert space ...
- 315
1
vote
2
answers
941
views
Alternate definitions of $C^{1,\alpha}$ and $C^{1,\alpha}(\bar{D})$ maps
My question is about the precise definition regarding the following:
Let $f$ be an orientation-preserving $C^1$ diffeomorphism of the unit circle $S^1$. So $f'(b)$ exists and can be thought as a ...
- 1,471
1
vote
1
answer
215
views
About principal values and Wirtinger derivative
Let $K$ be a compact of the plane of Lebesgues measure 0 and $\Omega$ a domain containing $K$. Denote by $E$ the vector space of functions that are holomorphic on $\Omega - K$.
I'm interested in ...
- 377
4
votes
2
answers
566
views
Which test functions are the divergence of a vector field?
The following apparently elementary question came out of a somewhat naive attempt to
prove that every distribution $u\in \mathscr D'(\mathbb R^2)$ with $\partial_1 u=\partial_2 u =0$ is a constant ...
- 16.5k
0
votes
1
answer
337
views
Integral inequality
Let $X$ be the d-dimensional hypercube $X=[0,1]^d$ and let $f$ and $g$ be such that $f(x) = 1$ if $x \in A$ and $0$ otherwise, $g(x)=1$ if $x \in B$ and $0$ otherwise, where $A$ and $B$ are generic ...
- 295
0
votes
1
answer
229
views
Complemented subspaces of $\ell_p(I)$ for uncountable $I$
I was looking for an article mimicing result of Pelczynski for $\ell_p$. I have found this one
Rodriguez-Salinas, B. (1994). On the Complemented Subspaces of $c_0(I)$ and $\ell_p(I)$ for $1 < p &...
- 1,697
3
votes
1
answer
199
views
Is P(X) a connected set for a set X with a $\sigma$-algebra P(X) and a measure function m on it to [0,$\infty$] when P(X) is equiped with meter d, that for every A,B in P(X), $d(A,B)=m(A \Delta B)$?
look at Problem14.12 of chapter3 of "Aliprantis-Burkinshaw-Principles of real analysis-3ed.1998" ; 12. Let A be the collection of all measurable subsets of X of finite measure. That is, A = {B in X: m(...
2
votes
1
answer
1k
views
Weierstrass factorization theorem in several variables
Can one indicate to me the Weierstrass factorization theorem in several variables (real or complex). In one complex variable the result is well known. Thank you in advance.
- 1,197
9
votes
3
answers
4k
views
Projections in Banach spaces
Dear All,
I am absolutely lost in the following problem:
Let $P_s, \: s \in [0,1],$ be a uniformly bounded family of projections (idempotents) in a Banach space $X$ such that $P_s P_t = P_{{\rm min}...
- 93
1
vote
1
answer
368
views
Is Every Symmetric Operator on the Schwartz Space Essentially Self-Adjoint?
More generally, suppose $S$ is a subspace of a Hilbert space $H$ that contains an orthonormal basis of $H$ (For example- the Schwartz space inside $L^2(\mathbb{R}^n)$). If $A:S \rightarrow S$ is ...
- 922
0
votes
1
answer
403
views
is the limit of ergodic functions still ergodic?
under what conditions is the limit of a sequence of ergodic functions still ergodic? are there simple counter-examples to this general statement?
- 21
0
votes
0
answers
231
views
Pure greedy algorithm
I study pure greedy algorithms in different basises. I am interested in 1 one question: is there such a Riesz basis $D$ in Hilbert space and $f\in H$ such that
$\|f-G_m(f,D)\|>Cm^{-1/2}\lvert\{f}\...
1
vote
0
answers
358
views
an infinite series expansion in terms of the polylogarithm function
we have the complex valued function :
$$f(z)=\sum_{n=0}^{\infty}a_{n}Li_{-n}(z)$$
we wish to recover the coefficients $a_{n}$ . the only thing i though would work is to try and come up with a function ...
- 181
1
vote
0
answers
289
views
Inequality regarding $\ell_p$ norms, $p<1$
Let $(x_{i,j})$ be an infinite double sequence of nonnegative real numbers, and $ 0< p<1$.
I would like to know whether one can bound from above the sum
\begin{equation}
\sum_{i,j} x_{i,j}^p
\...
- 61
1
vote
1
answer
298
views
Maximal spectrum of a complex, unital and commutative Banach-algebra
Let $A$ be a complex, unital and commutative Banach-algebra.
Question: Is the maximal spectrum $Max(A)$ of $A$ endowed with the topology induced by the prime spectrum $Spec(A)$ of $A$, Hausdorff?
...
- 328
4
votes
0
answers
140
views
When is $A^*A$ invertible for Banach space?
Let's consider a linear functional $A$ from smooth objects to smooth ones. It is first order operator in the sense that it extends to be a map from $W^{k+1,p}$ to $W^{k,p}$. Assume that we have $L^2$ ...
- 527
0
votes
2
answers
160
views
Bounded inverse to morphism of Banach algebras
Let $A:X\to Y$ be a surjective morphism of Banach spaces.
1) Does there always exists $B_R$, a bounded right inverse to $A$?
2) Assume additionally that $A$ is a morphism of unital Banach algebras. ...
- 580
2
votes
1
answer
132
views
Form of finite dimensional contractive projection in $L_p$
Let $P$ be a finite dimensional contractive (norm 1) projection in $L_p$, $1 < p < \infty$. Then $P$ is of the following form:
$Pf = \sum_{k=1}^n g_k \int h_kf$
Where $\|g_k\|_p = \|h_k\|_q = \...
- 311
8
votes
2
answers
464
views
Direct proof of "K is projective iff C(K) has the Hahn-Banach property" ?
An object $X$ of a given category is called projective if for each morphism $f : X \rightarrow Z$, and each epimorphism $ g : Y \twoheadrightarrow Z$, there is a morphism $h : X \rightarrow Y$ such ...
- 7,249
6
votes
6
answers
1k
views
Proving continuity on spaces of distributions?
Let $\mathcal{D}'(\Omega)$ be the space of distributions on an open set $\Omega$, and $\mathcal{E}'(\Omega)$ the compactly supported ones.
When you have a linear operator $T:\mathcal{D}'(\Omega)\...
- 61
1
vote
1
answer
142
views
Linear Maps between $L^1$-spaces of singular measures
I posted the following question also here, but thought that I can get more answers in MO.
Let $(\Omega,\Sigma)$ be a measurable space and $\nu_1$, $\nu_2$ two probability measures on it. For $i=1,2$, ...
- 13
4
votes
1
answer
615
views
Isometric embeddings of $\ell_q^m$ into $\ell_p$ and $L_p$ for $p,q\in[1,+\infty]$
I'm looking for articles describing or proving nonexistence of isometric embeddings of $m$-dimensional space $\ell_q^m$ into $L_p$ and $\ell_p$ for $q,p\in[1,+\infty]$.
Since $\ell_q^m$ is finite ...
- 1,697
6
votes
0
answers
369
views
Paving conjecture for Toeplitz matrices
Let me first recall what is the so-called paving conjecture:
for any $\epsilon >0$, there exists $r\in \mathbb N$ such that
for any bounded operator $A$ on $\ell^2(\mathbb Z)$, there exists a ...
- 16.2k
2
votes
2
answers
2k
views
Does the Fourier series of an $L^1$ function converge to the function *weakly* in $L^1$?
Let $f$ be a periodic $L^1$ function, and $S_n[f]$ the $n$-th partial sum of its Fourier series. I am aware that $S_n[f]$ might not converge toward $f$ in $L^1$ (i.e., in norm). However, does it at ...
- 32.5k
0
votes
0
answers
244
views
Checking whether this would be bounded
It may be better to post this question here. Assume that $M$ is an $m$ by $m$ ($m$ is an even number) symmetric
positive-semi-definite matrix with exactly $m/2$ positive eigenvalues
and every entry of ...
- 1
2
votes
1
answer
637
views
Partial order on self-adjoint extensions?
Is there a natural partial order and/or lattice structure on the set of closed symmetric or self-adjoint extensions of a densely defined, unbounded, symmetric operator on a Hilbert space? Any ...
- 21.6k