All Questions
9,958 questions
11
votes
0
answers
1k
views
Is the Fourier-Transform a bounded operator on Lorentz spaces L(2,q)?
It is well known that the Fourier transform $\mathcal{F}$ maps $L^1(\mathbb{R}^n)$ continuously into $L^\infty(\mathbb{R}^n)$ and $L^2(\mathbb{R}^n)$ continuously into $L^2(\mathbb{R}^n)$.
Then, by ...
- 111
3
votes
1
answer
280
views
An analogue of an old proposition
For the absolute value $|C|=(C^*C)^\frac{1}{2}$ and the
Hilbert-Schmidt norm
$\parallel C\parallel_{HS}=(trC^*C)^\frac{1}{2}$ of the operator $C$. The
following inequality is shown by Araki et al in ...
- 25.8k
8
votes
2
answers
915
views
Group homomorphisms and maps between function spaces
Let G and H be locally compact groups, and let $\theta:G\rightarrow H$ be a continuous group homomorphism. This induces a *-homomorphism $\pi:C^b(H) \rightarrow C^b(G)$ between the spaces of bounded ...
- 18.7k
8
votes
1
answer
713
views
Factoring operators $L_\infty \longrightarrow L_2$ as the composition of $n$ strictly singular operators, $n\in \mathbb{N}$
Motivation and background This question is motivated by the problem of classifying the (two-sided) closed ideals of the Banach algebra $\mathcal{B}(L_\infty)$ of all (bounded, linear) operators on $L_\...
- 31.5k
15
votes
4
answers
2k
views
Naive questions about "matrices" representing endomorphisms of Hilbert spaces.
This is a very basic question and might be way too easy for MO. I am learning analysis in a very backwards way. This is a question about complex Hilbert spaces but here's how I came to it: I have in ...
- 75
4
votes
1
answer
311
views
Continuous functions on the states of a C*-algebra and its elements
Let $\mathcal A$ be a C*-algebra and $s(\mathcal A)$ the set of states on $\mathcal A$, with the weak* topology, as a subspace of the dual space. Suppose $f: s(\mathcal A) \to \mathbb C$ is a ...
- 9,366
13
votes
5
answers
1k
views
Does this sequence span $L^2$?
Consider the following sequence of functions in $L^2[0,\infty)$:
$$f_n(x)=e^{-x/n}x^n,\;\;n\geq 1$$
Does this sequence span $L^2[0,\infty)$ (that is, is the set of finite linear combinations
of these ...
- 1,471
12
votes
3
answers
1k
views
Drawing conclusions by NOT using AC.
The existence of non-measurable subsets and functions on $\mathbb{R}$ require the use of the axiom of choice. That is, there exist models of ZF in which all subsets of (and hence all functions defined ...
- 3,631
2
votes
2
answers
679
views
L^2 space of holomorphic functions with given weight
Hi folks, what is known about the $L^2$ space of holomorphic functions of 1 complex variable with the scalar product
$\langle f, g \rangle = \int dzd{\bar z} \frac{ {\bar f(z)} g(z) }{(1 + z{\bar z})^...
- 3,183
2
votes
1
answer
272
views
Contractions and spaces
Suppose $X$ is a closed subspace of an $L^{1}$-space and $X$ is isometric to another $L^{1}$-space. Then we know that $X$ is in the range of a contractive projection on the $L^{1}$-space. Is there any ...
- 25.8k
4
votes
1
answer
466
views
Injection between non-isomorphic irreducible Hilbert space reps?
I must be missing something trivial here.
Let $G$ be, say, a reductive Lie group (or more generally any locally compact Hausdorff unimodular topological group). A unitary Hilbert space representation ...
- 4,624
10
votes
5
answers
1k
views
What is a rigorous statement for "linear time-invariant systems can be represented as convolutions"?
In Signal Processing books, a fundamental theorem is that linear time invariant systems can be represented as a convolution with a distribution. Could you give a mathematically rigorous statement of ...
- 2,745
3
votes
3
answers
1k
views
Minimizing a functional
I have wondered the problem in http://www.helsinki.fi/~hmkokko/Stuff/Esdale/index.html for over year without success. If we try to minimize the functional equation
T(\theta ) = \int_0^L\frac {dx}{v_0\...
- 29k
1
vote
1
answer
1k
views
Besicovitch Covering Constant for R^1
In the case where $E\subset\mathbb{R}^1$, a Besicovitch cover of $E$ is a cover by open intervals such that each point of $E$ is the center of some interval in the cover.
The Besicovitch Covering ...
- 320
1
vote
1
answer
433
views
Intersection of ideals in C*-algebra or even rings in general
Dear all,
here is a question that has been bothering me. It goes without saying that I would appreciate any help in answering it.
Let {I_k} be a countable sequence of two sided closed ideals in a C*-...
- 20.8k
1
vote
0
answers
133
views
Square powers of hemicontinuous operators
Let H be an infinite dimensional real Hilbert space.
A [not necessarily linear] mapping of H into itself is said to be hemicontinuous if it is continuous from each line
segment of H to the weak ...
- 4,060
2
votes
2
answers
768
views
Elementary vector measure question: what am I doing wrong?
This is an edited post of a post I made on sci.math (e.g. to fit MO markup) with
an elementary question on vector measures. Since it is almost a week and I have
received no answers, I am trying here. ...
- 11.4k
3
votes
1
answer
556
views
"Radon-Nikodym theorem" for nonabsolute continuous measures
Recently, in a particular problem I was solving, I needed some kind of Radon-Nikodym theorem for measures where one of them is not necessarily absolutely continuous with respect to other.
My colleague ...
- 41.1k
94
votes
1
answer
11k
views
The mathematical theory of Feynman integrals
It is well known that Feynman integrals are one of the tools that physicists have and mathematicians haven't, sadly.
Arguably, they are the most important such tool. Briefly, the question I'd like to ...
- 2,097
10
votes
2
answers
1k
views
Are operators with trivial spectrum nilpotent in a sense?
Being far from analysis, I recently learned about the Invariant subspace problem and came up with the following (perhaps simple or well-known) question.
Let $H$ be a separable complex Hilbert space ...
- 60.5k
7
votes
1
answer
1k
views
Banach spaces with a certain separability property
In Ledoux and Talagrand's "Probability in Banach Spaces", for technical reasons they frequently assume that a Banach space $B$ has the property that the unit ball of $B^*$ contains a countable subset $...
- 1,544
1
vote
2
answers
3k
views
unit sphere is weak dense in the unit ball
As I remember the following is true:
Fact: for every infinite-dimensional normed space $X$
the unit sphere $S$ is weak-dense in the unit ball $B$.
Please help me find a reference.
Thanks in ...
- 29.8k
11
votes
2
answers
2k
views
What's wrong with compact-open topology on the space of maps?
Given a smooth vector bundle $E$ with non-compact base, let
$\Gamma(E)$ be the space of $C^\infty$ sections equipped with compact-open $C^\infty$-topology.
I have heard that $\Gamma(E)$ is not ...
- 26.8k
8
votes
1
answer
1k
views
Borel(X) = \sigma(X') for X non-separable
Let $X$ be a Banach space, $X' = \mathcal{L}(X, \mathbb{K})$ its dual space. Denote by $\mathcal{B}(X)$ the $\sigma$-algebra of Borel sets and denote by $\sigma(X')$ the $\sigma$-algebra which is ...
- 41.1k
6
votes
2
answers
1k
views
Quantitative questions about the size of a finite epsilon net
Let $X$ be a metric space, and let $U \subset X$ be any set. A finite set $N = N(\epsilon) \subset U$ is called a finite $\epsilon$-net of $U$ if every point of $U$ is at most a distance of $\epsilon$...
- 943
4
votes
1
answer
985
views
weak convergence in infinite dimensional spaces
Weak convergence can be tricky when dealing with infinite dimensional spaces. For example, the usual Levy's continuity theorem does not extend readily to separable Banach spaces.
Consider a (...
- 3,069
6
votes
0
answers
2k
views
Weak lower semi-continuity
Which conditions assure the weak lower semicontinuity of, say, an integral functional of the type
$F(u):=\int_\Omega f(u(x),Du(x))dx$ on $W^{1,2}(\Omega,\mathbb{R}^N)$ for a bounded, if you will even ...
9
votes
5
answers
870
views
Abelianization of GL(H)
This is related to Theo's question about the abelianizations of finite dimensionsal Lie groups.
I am interested in a specific (infinite-dimensional) case of the above question. Let H be an infinite-...
CommunityBot
- 1
11
votes
1
answer
654
views
Nonseparable Hilbert spaces as quotients of spaces of bounded functions
Is the following result true: the Hilbert space $\ell^{2}\left(2^{\Gamma}\right)$ is a quotient of $\ell^{\infty}\left(\Gamma\right)$ for any
uncountable $\Gamma$ ? [I think it is, but cannot remember ...
- 31.5k
7
votes
2
answers
808
views
Is a subspace with a certain property dense in the dual of a vector space?
Suppose we have a normed vector space $V$ and its dual $V^*$, and suppose that $X \subseteq V^*$ has the property that for every $v \in V$, there is some $\phi \in X$ with $\Vert \phi \Vert = 1$ such ...
- 4,060
0
votes
1
answer
635
views
Topological dual and the notions of "smaller" and "larger" than...
Hi,
I've read this sentence but I can not understand what it means
[...] $\Phi'$ is the topological dual of some dense space $\Phi$ of $H_{aux}$ [...] Notice that the choice of $\Phi$ is subject to ...
- 39k
2
votes
2
answers
354
views
A bound on linear functionals over cotype 2 spaces
This is a modification of the somewhat naive question that I asked below.
Suppose $X$ is a real Banach space of cotype-2, and $u_1, u_2, ... u_n$ are unit vectors in this space. For $\gamma = ((\...
- 31.5k
7
votes
1
answer
347
views
Nonexistence of determinantal functional equation for $\arccos$
Suppose I have distinct real numbers $a_i \in [-1,1]$, $i \in [k]$. I want to choose real numbers $b_j, j\in [k]$ such that the matrix $(\arccos(a_i b_j))_{i,j \in [k]}$ is nonsingular.
Is this ...
- 32.4k
5
votes
1
answer
403
views
Local form of a real-analytic function taking values in a Banach space
Let $B$ be an infinite-dimensional Banach space, and let $M\subset\mathbb{R}^n$ be a neighborhood of the origin in $\mathbb{R}^n$.
Suppose that $I:M\to B$ is a real-analytic function with $I(0)=0$ ...
- 9,366
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). ...
CommunityBot
- 1
3
votes
2
answers
766
views
Borel vs measure for all Borel measures
Let X be locally compact and Hausdorff, and let $f:X\rightarrow\mathbb R$ be a function. Suppose that for all finite regular (positive) Borel measures $\mu$, we know that $f$ is $\mu$-measurable. ...
- 41.1k
11
votes
1
answer
2k
views
Algebraic properties of the algebra of continuous functions on a manifold.
Does the algebra of continuous
functions from a compact manifold to
$\mathbb{C}$ satisfy any specific
algebraic property?
I'm not sure what kind of algebraic property I expect, but I feel that ...
- 7,329
11
votes
0
answers
657
views
For which Lie groups is the convolution of any two nonzero integrable compactly supported functions nonzero?
The Titchmarsh convolution theorem implies that the convolution of two nonzero functions $f,g\in L^1(\mathbb R)$ with compact support is nonzero. There is a generalization of this theorem to the case ...
- 637
2
votes
3
answers
946
views
How can I measure the Morse index in infinite dimensions?
Let $V$ be a vector space over $\mathbb R$, and $a: V\otimes V\to \mathbb R$ a symmetric bilinear pairing. Recall that the Morse index of $a$ is the maximal dimension of any subspace $V_- \subseteq V$...
CommunityBot
- 1
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 $A$, ...
CommunityBot
- 1
1
vote
3
answers
5k
views
rules for operator commutativity?
Hi, my apologies for a rather non-specific question. I wonder if there is a general set of conditions under which operators are commutative in functional analysis. Most that I've found is that "...
- 41.1k
1
vote
0
answers
660
views
Fractional Fourier transform [closed]
Let $T: L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n)$ be the Fourier transform. Is there any reasonable definition of fractional Fourier transform (i.e. operator $A$ such that $A^{\alpha}=T$ for $\...
- 3,323
12
votes
3
answers
646
views
Radii and centers in Banach spaces
Suppose I have a Banach space $V$ and a set $A \subseteq V$ such that for all $\epsilon > 0$ there exists $v$ such that $A \subseteq \overline{B}(v, r + \epsilon)$. Does there exist $c$ such that $...
- 4,060
2
votes
0
answers
197
views
Generating cones having no surjections [in operator spaces]
Is this little toy known ?
Let $E$ be some Banach space, and let $K$ be the closed unit ball
of its dual, endowed with the weak-star topology. Also, let $j:E$ $\rightarrow$ $C(K)$
be the natural ...
- 25.8k
1
vote
1
answer
359
views
Convergence of operators to the identity on Banach spaces
Let $U_\infty$ be a compact space, and let $U_r$ be an increasing family of compact subspaces whose closure is all of $U_\infty$. That is, $U_r \subseteq U_{r'}$ if $r \le r'$ and $U_\infty = \...
- 25.8k
12
votes
4
answers
1k
views
Topologizing free abelian groups
For any set $S$ one can consider the free abelian group $\mathbb{Z}[S]$ generated by this set. Now suppose, there is a topology on $S$ given. Is it possible to find a topology on $\mathbb{Z}[S]$ in ...
- 63.1k
4
votes
1
answer
1k
views
When can a partial isometry $u$ in $\mathcal B(H \otimes K)$ be extended to a unitary in $1 \otimes \mathcal B(K)$?
Let $H$ and $K$ be Hilbert spaces, and let $u$ be a partial isometry in $\mathcal{B}(H \otimes K)$ between projections $p_0 = u^\ast u$ and $p_1 = u u^\ast$ such that $p_0, p_1 \leq 1 \otimes (1-q)$ ...
- 871
1
vote
1
answer
2k
views
spectra of sums and products in (Banach) algebras [was: Spectrum in Banach Algebra]
Let a,b be 2 elements in a Banach Algebra.Let Spec(x) denote the spectrum of an element x. If a,b commute with each other, then by Gelfand Transformation, we have Spec(a+b) is a subset of Spec(a)+Spec(...
- 7,329
1
vote
1
answer
994
views
On the convolution of generalized functions
It is provable that $f_\lambda\to f\Rightarrow f_\lambda*g\to f*g$ if $g$ has a compact support (shown in my textbook). In my particular case, $g=u(t+\triangle t)-u(t-\triangle t)$. Does for that ...
- 26.8k
2
votes
2
answers
584
views
A proof about an unconditional basis theorem
Hello everyone. I'm in a little trouble trying to find the proof of a theorem stated by W. T. Gowers. It is the Lemma 1.6 in his article 'An infinite Ramsey theorem and some Banach space dichotomies' (...
- 31.5k