All Questions
10,049 questions
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Density of Dolean exponentials in L2 and Wiener Measure
Assume that W is the classical Wiener space C([0,1],R) note $\mu$ the Wiener measure, and denote by $\mu_s$ the image of $\mu$ under the maping $T: W ->W$ such that$ T(w)= \sqrt(s) w$ . Denote by $...
- 19
8
votes
1
answer
612
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Is the set of exponentials open?
Let $A$ be a $C^*$-algebra or some norm-closed algebra of operators on a Hilbert space.
In the old paper
Hille, E. On Roots and Logarithms of Elements of a Complex Banach Algebra, Math. Annalen, ...
- 25.5k
-1
votes
1
answer
311
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A differential equation
let $g(s)$ be real-valued function defined on $[0,T]$ such that $g(T)=0$ and suppose that $g$ is a "nice function"
Assume that $0<\gamma<1$, $v$ is a positive number, and
$$\frac{dg}{ds}+(v\...
- 1
1
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1
answer
706
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Plancherel-Polya Type Inequality for non-compactly Fourier-supported Functions??
Hi!
The Plancerel-Polya inequality can be stated as follows:
Let $0 < p\le \infty$ and $ \nu \in \mathbb{Z}$. Suppose that $g$ is a (smooth) function satisfying $\mbox{supp }\hat g \subset \...
- 979
2
votes
1
answer
1k
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Hilbert Schmidt operators
I don't know much about the theory of Hilbert spaces but a research project has me working with them a little bit. In particular requiring an operator to be Hilbert-Schmidt is a recurring condition. ...
- 3,968
7
votes
3
answers
1k
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Nice orthonormal basis for L^2(Cantor set)
Let X be the Cantor set, which we view as the space $2^\mathbb{N}$ (the set of all infinite binary sequences), equipped with the product topology. We can construct a Borel probability measure $\mu$ on ...
- 81
4
votes
2
answers
707
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Selecting basic sequences
Suppose $(x_\alpha)_\alpha$ is an uncountable, linearly independent family of norm one vectors in a Banach space. Can one always select a basic sequence (or at least a minimal system) from this family?...
- 355
3
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1
answer
2k
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Does the category of Hilbert spaces possess a product?
I've been studying some category theory lately and in particular, I became acquainted with the notions of products and coproducts, which led me to ponder the following:
Consider the category of all ...
- 4,874
5
votes
1
answer
666
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Question regarding divergence
Let $E$ be a closed and convex set of distributions on a finite set $A$. Let $P',Q'\notin E$ and let $P^{\star},Q^{\star}$ be their respective estimates in $E$ with respect to the KL-divergence, i.e.,...
- 779
1
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1
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635
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Closed range for a continuous linear transformation
I have a Banach space $B$ and a continuous linear transformation $F:B \rightarrow B\times B$. One of the induced transformations $F(1):B \rightarrow B$ and $F(2):B \rightarrow B$ into the factors of ...
- 549
7
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1
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737
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Question about projections on a Hilbert space
Sorry for the vague title, I can't think of a better one that isn't overly long.
Suppose that $S$ is a commuting set of projection operators on a Hilbert space. I'll introduce the following notation: ...
- 391
7
votes
2
answers
790
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Question about von Neumann algebra generated by a complete algebra of projections
Hi all, sorry if this is a dumb question, I don't know much about von Neumann algebras except the definition and a few relevant facts I've managed to prove by myself so I expect the answer will turn ...
- 391
1
vote
1
answer
717
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Double dual space of a C* algebra A
We know that $A$ embeds into $A$** (the double dual space of $A$ ). Is the following true? If $\Psi$ is in $A$** and weak* continuous, is there an element $a \in A$ such that $ \Psi$ is the ...
- 831
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920
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Looking for references talking about category of topological vector spaces
It's known that category of topological vector spaces is not abelian but quasi-abelian or exact category. I am looking for the references playing with this category(category theory). All the related ...
- 5,515
5
votes
2
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579
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Improved versions of discontinuous functions
Given a set X (such as the set of points in an interval), the space ℝX of all real-valued functions on X is not usually the function space we work with -- it is "too large" in some sense. Thus, ...
- 8,903
21
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0
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876
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Are the eigenvalues of the Laplacian of a generic Kähler metric simple?
It is a theorem of Uhlenbeck that for a generic Riemannian metric, the Laplacian acting on functions has simple eigenvalues, i.e., all the eigenspaces are 1-dimensional. (Here "generic" means the set ...
- 6,247
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3
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1k
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If *Y* is weakly dense in *X*, is the unit ball in *Y* necessarily dense in the unit ball in *X*?
Let X be a normed space and denote by X* the space of all bounded linear functionals on X. Take a linear subspace G ≤ X* which separates the elements of X, i.e., for each x ∈ X, there is an f &...
- 73
3
votes
1
answer
1k
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Cyl(E) = Borel(E) for E non-reflexive Grothendieck Banach space
This is sort of a follow-up to Borel(X) = \sigma(X') for X non-separable
PROBLEM: Given a Banach space $E$ over $\mathbb{K} \in \{\mathbb{C}, \mathbb{R}\}$ that has the Grothendieck property. ...
- 1,338
2
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1
answer
2k
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Cesaro convergence implies weak convergence of a subsequence
Suppose a bounded sequence $(x_n)$ converges to $x$ in the Cesaro sense (i.e., $\frac{1}{n}(x_1 + x_2 + \dots + x_n)\rightarrow x$) in a separable Hilbert space $H$. How to prove that some subsequence ...
- 5,019
1
vote
2
answers
3k
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Weak-* compactness in L^1
Hey I'm really stuck on what I think is an interesting 'paradox'. Consider the sequence of functions $f_n = 1_{[n,n+1]}$ (indicator functions of the interval $[n,n+1]$.
These are uniformly bounded ...
- 19
3
votes
1
answer
615
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When is a fixed point of f^n a fixed point of f?
Let $E$ be a Banach space and $f:E\to E$ be a continuous map. By $f^n$ we denote the $n$-th iterate of $f$, i.e. $f^n:=\underbrace{f\circ f\circ\cdots \circ f}_{\text{n times}}$. Let $x_0$ denote a ...
- 41
11
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1
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603
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Reference for a particular Radon transform on non-positively curved spaces
Let me first recall that the classical Radon transform takes a (smooth compactly supported, say) function $f$ defined on $\mathbb{R}^n$ as an input, and gives as output the map $H\mapsto \int_H f$ for ...
- 14.4k
14
votes
1
answer
1k
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Any further applications of Freudenthal's 1936 Spectral Theorem?
Seemingly completely forgotten, back in 1936, the Dutch mathematician Freudenthal, quite well known at the time, proved his so called Spectral Theorem, see chapter 6 in Luxemburg & Zaanen : Riesz ...
5
votes
2
answers
1k
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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
views
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
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1
answer
3k
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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
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1
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577
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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
5k
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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
3
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
10
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2
answers
960
views
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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228
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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
874
views
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
5
votes
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 $\phi_h(t)=\dfrac{\phi(t+h)-\phi(t)}{h}$...
- 3,184
4
votes
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
views
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
votes
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
8k
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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
votes
2
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674
views
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
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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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0
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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
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1
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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
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
25
votes
1
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8k
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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
2
votes
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
votes
1
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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
votes
1
answer
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
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24k
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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 ...