All Questions
Tagged with functional-analysis or fa.functional-analysis
9,773 questions
5
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Applications of minmax theorem(s)
Intro We suppose $X$ and $Y$ are nonempty sets and f: $X\times Y \rightarrow \mathbb{R}$. A minimax theorem is a theorem that asserts that, under certain conditions,
$$ \inf_Y \sup_X f = \sup_X \...
6
votes
1
answer
444
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When does a matrix define a convolution operator on a hypergroup?
Let $H$ be a discrete hypergroup. Suppose I have a matrix $A=(A_{x,y})$ indexed over $H$ with nonnegative entries which defines a bounded operator on $\ell^2(H)$. When does there exist $f\in\ell^1(H)$ ...
- 5,425
18
votes
1
answer
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Let a function f have all moments zero. What conditions force f to be identically zero?
Throughout, let $f$ be a Lebesgue measurable function (or continuous if you wish, but this is probably no easier). (Questions with distributions etc. are possible also but I want to keep things simple ...
- 1,990
7
votes
1
answer
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Does a crossed product R⋊_α F_n of the hyperfinite factor of type II_1 and a free group have the QWEP?
Let $\mathcal{R}$ be the hyperfinite factor of type $\rm{II}_1$ and let $\mathbb{F}_n$ be a free group with $n$ generators. Let $\alpha$ be an action of $\mathbb{F}_n$ on $\mathcal{R}$.
Does the von ...
- 1,222
18
votes
1
answer
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Unbounded linear operator defined on $l^2$
Let $l^2$ be a Hilbert space of infinite sequences $(z_0, z_1, \cdots)$ with finite $\sum_{i=0}^{\infty} |z_i|^2$.
Are there any simple example of unbounded linear opearator $T: l^2 \to l^2$ with $D(...
- 2,821
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votes
1
answer
1k
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Adjoint/transpose of wavelet transform
I'm using a wavelet transform in Matlab, so I think of it as a black-box. I'll represent it here as $W(x)$. There's a reconstruction function as well, which I'll write as $W^\dagger(y)$. I can ...
- 170
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2
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959
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Stone-Weierstrass for cones
A version of the Stone-Weierstrass Theorem asserts: If A is a linear subspace of C(K), the set of continuous functions on a compact space, and if A is a subalgebra that contains the constant functions ...
- 101
4
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1
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When can closedness of the range of an operator be checked on a positive cone?
Let $T:X\to Y$ be an operator between Banach spaces $X$ and $Y$. Assume that $X$ has a positive cone $C\subset X$, which generates $X$: every element of $X$ can be written as a difference of elements ...
user2412
2
votes
2
answers
873
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Dimension of the space of harmonic functions on the unit ball
Is the dimension of the space of $H^2(B)$ harmonic functions on unit ball $B\subset\mathbb{R}^d$ countably or uncountably infinite?
- 21
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2
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3k
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Uniform convergence of difference quotient
Let $\phi\in C^\infty_c(\mathbb R)$ be a smooth function with compact support.
For $h>0$ define the difference quotient $\phi_h\in C^\infty_c(\mathbb R)$ by $\phi_h(t)=\dfrac{\phi(t+h)-\phi(t)}{h}$...
- 3,174
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3
answers
3k
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Distributional derivative of non continuously differentiable functions
Hello,
let $f$ be a continuously differentiable function on $R^n$. Then its classical derivative and its distributional derivative coincide.
It is known (cf. Rudin, Functional Analysis, Sect. 6.13) ...
- 5,327
6
votes
1
answer
581
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A puzzling question on real interpolation
Suppose an operator $T$ is bounded on $L^2$ and also bounded from $L^{1}$ to $L^{1}$-weak. Then by Marcinkewicz interpolation one gets that $T$ is bounded on every $L^{p}$ for p between 1 and 2. ...
- 9,007
3
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2
answers
1k
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Spectral decomposition for an arbitrary linear combination of position and momentum operators
Suppose we have the Hilbert space L2(Rn) and we have n operators Qi and n operators Pi defined in the usual way by:
Qi ψ(q1,q2,...,qn) = qi ψ(q1,q2,...,qn)
Pi ψ(q1,q2,...,qn) = -i $\frac{...
- 195
3
votes
6
answers
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Functional Analysis and its relation to mechanics
Hi I'm currently learning Hamiltonian and Lagrangian Mechanics (which I think also encompasses the calculus of variations) and I've also grown interested in functional analysis. I'm wondering if there ...
- 33
9
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2
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674
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Small crown probabilities (and infinite dimensional margin assumption)
My question is:
How do I find sharp upper bounds on $P(|q|\leq \epsilon)$ uniformly over a set of gaussian polynomes $q$ of degree two.
Notations and definitions (to make the question rigorous)
Let ...
23
votes
3
answers
6k
views
Density of smooth functions under "Hölder metric"
This question came up when I was doing some reading into convolution squares of singular measures. Recall a function $f$ on the torus $T = [-1/2,1/2]$ is said to be $\alpha$-Hölder (for $0 < \alpha ...
- 505
1
vote
0
answers
308
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Loynes spaces, also called pseudo-Hilbert spaces
Let me first define my object:
First, a locally convex space $Z$ is called admissible in the sense of Loynes if
$Z$ is complete
There is a closed convex cone in $Z$, called $Z_+$, satisfying (for $x\...
- 2,600
26
votes
3
answers
11k
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L1 distance between gaussian measures
L1 distance between gaussian measures: Definition
Let $P_1$ and $P_0$ be two gaussian measures on $\mathbb{R}^p$ with respective "mean,Variance" $m_1,C_1$ and $m_0,C_0$ (I assume matrices have full ...
- 833
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1k
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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
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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 ...
- 223
8
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2
answers
915
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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
25
votes
1
answer
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Convergence of Fourier Series of $L^1$ Functions
I recently learned of the result by Carleson and Hunt (1968) which states that if $f \in L^p$ for $p > 1$, then the Fourier series of $f$ converges to $f$ pointwise-a.e. Also, Wikipedia informs me ...
- 871
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votes
6
answers
8k
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Unbounded operator bounded in a dense subset
Let $X, Y$ be normed vector spaces, where $X$ is infinite dimensional. Does there exist a linear map $T : X \rightarrow Y$ and a subset $D$ of $X$ such that $D$ is dense in $X$, $T$ is bounded in $D$ (...
- 783
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4
answers
1k
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An inequality question
Let $M$ be a $3\times2$ matrix. Is it true that for any $x\in\mathbb{R}^{2}$
with $\left\Vert x\right\Vert _{3}=1$ there is some subspace $V$
with dimension $2$ of $\mathbb{R}^{3}$, such that $\left\...
- 59
8
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1
answer
713
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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_\...
- 2,378
4
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1
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311
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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 ...
- 41
152
votes
18
answers
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Why do we care about $L^p$ spaces besides $p = 1$, $p = 2$, and $p = \infty$?
I was helping a student study for a functional analysis exam and the question came up as to when, in practice, one needs to consider the Banach space $L^p$ for some value of $p$ other than the obvious ...
5
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680
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Does the norm of a normed linear space determine the form of its dual spaces elements?
Hello everybody,
As an introductory example, suppose $U \subset R^n$ is open and bounded, let $p = 2$. Then there is a constant $c>0$ s.t. $\forall u \in W^{1,p}_0 : \Vert u \Vert _ {W^{1,p}_0} \...
- 5,327
2
votes
2
answers
679
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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})^...
- 362
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5
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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 ...
- 520
2
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1
answer
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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 ...
- 95
4
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466
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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 ...
- 41.4k
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3
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When does a unitary Hilbert space rep of a reductive Lie group decompose into a direct sum of irreps with finite multiplicities?
I'm giving some lectures on the trace formula. Here's something I proved in the last lecture. Let $G$ be a locally compact Hausdorff unimodular topological group (e.g. a reductive Lie group), let $\...
- 41.4k
81
votes
3
answers
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Norms of commutators
If an $n$ by $n$ complex matrix $A$ has trace zero, then it is a commutator, which means that there are $n$ by $n$ matrices $B$ and $C$ so that $A= BC-CB$. What is the order of the best constant $\...
- 31.5k
4
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1
answer
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Existence of weak limits
Background: Three months ago, I asked this question, which is a bit related to the following (if the answer to it was Yes, then answer to this one would be Yes too, but since that was a No, it still ...
27
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1
answer
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Criteria for boundedness of power series
Consider a power series $\sum_{n=0}^{\infty} a_n x^n$ that is convergent for all real
x, thus defining a function $f: \mathbb{R} \to \mathbb{R}$.
Can one give necessary and sufficient criteria the ...
- 4,014
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7
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Why is the Hahn-Banach theorem so important?
Every time I hear it mentioned it is praised in the highest possible terms, and I remember one of my old lecturers saying that it is one of the 3 most important theorems in analysis. Yet the only ...
- 4,351
1
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1
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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 ...
- 111
1
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1
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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*-...
7
votes
3
answers
495
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Noninteger iterates of functions: How to get ODE from flow at a given time?
Suppose you have the autonomous ordinary differential equation $dx(t)/dt = f(x(t))$ with $x: \mathbb{R} \to \mathbb{R}$ and the initial condition $x(0)=x_0$. Then, assuming some regularity conditions, ...
- 4,014
4
votes
2
answers
4k
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Proof of Young's convolutions inequality for a general measure on $\mathbb R^d$
Is Young's inequality true for an arbitrary measure on $\mathbb R^d$? If so, where can I find a proof of it? In particular, where can I find the proof of the discrete version (i.e the version for $\...
- 2,745
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3
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Relevance of the complex structure of a function algebra for capturing the topology on a space.
This question is the outcome of a few naive thoughts, without reading the proof of Gelfand-Neumark theorem.
Given a compact Hausdorff space $X$, the algebra of complex continuous functions on it is ...
- 3,699
15
votes
4
answers
2k
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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 ...
- 41.4k
7
votes
3
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Can Stein's maximal principle be strengthened?
Let $T$ be an operator on $S(G)$ where $G$ is the line $R$ or the circle $T$, and $S(G)$ denotes the Schwartz space of functions on $G$.
We can ask if the operator T is bounded (as an operator from $...
- 13k
22
votes
2
answers
2k
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Examples of loss of regularity by "creation of topology"
I would like to have a list as general as possible of examples of situations where the density of smooth objects into some "natural class" (the meaning of "natural" depending on the problem considered)...
- 2,041
21
votes
5
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18k
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When is Sobolev space a subset of the continuous functions?
If we let $\Omega\subset\mathbb{R}^d$ with $d=1,2,3$ and define $\mathcal{H}^1(\Omega)=(w\in L_2(\Omega): \frac{\partial w}{\partial x_i}\in L_2(\Omega), i=1,...,d)$. My tutor has repeated several ...
- 3,217
1
vote
0
answers
133
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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
5
votes
1
answer
7k
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Dual Spaces of Sobolev Spaces
I will consider Sobolev spaces with $p=2$, only, so that they are Hilbert spaces. Hence the Sobolev inner product identifies each Sobolev space with its dual. In other words, I have an isomorphism $H^...
- 807
2
votes
2
answers
768
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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. ...
- 1,848
3
votes
1
answer
556
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"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 ...
- 349