All Questions
9,958 questions
1
vote
2
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733
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Quantum Error Correction
One can correct the errors in a quantum channel iff the coherent information of the input state is not reduced by the channel. This is analogous to sending quantum entanglement through a channel. If ...
- 6,342
4
votes
0
answers
257
views
A matrix minimisation problem
Feel free to edit the title!
Suppose A is a C*-algebra and $a,b\in A$ are self-adjoint. I'd be very happy with A being just $n\times n$ matrices.
Question: If there are $t\in\mathbb R$ and $\...
- 39k
7
votes
1
answer
481
views
Gelfand theory Problem
I have 2 problems in Gelfand theory. I shall be thankful for any
answers.
1)What is the gelfand spectrum of l^1(N)?
A few of
the elements are evaluations of functions(defined below) on closed
unit ...
- 25.8k
2
votes
1
answer
412
views
General Sobolev Inequalities
In Partial Differential Equation by Lawerence Evan p284 there is this theorem stated:
Let $U$ be a bounded open subset of $\mathbb{R}^n$ with $C^1$ boundary. Suppose $u\in W^{k,p}$ then if $k>n/p$ ...
- 52.3k
1
vote
1
answer
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Square root of integral operator
Consider the 1-torus $\mathbb{T}$. Let $k$ be a smooth function on $\mathbb{T}^2$ and $K$ be the integral operator on $L^2(\mathbb{T})$ with kernel $k$. One can show that $K$ is of trace class, hence $...
- 2,793
1
vote
2
answers
606
views
Do separable $C^*$-algebras form a set?
The question is in subject.
Update: See Andreas Thom's answer.
- 16.2k
0
votes
2
answers
2k
views
fundamental solution of radial wave equation
i am trying to find resources on the derivation of the fundamental solution to the radial wave equation. any suggestions of or links to books, papers, and/or notes would be much appreciated. i have ...
CommunityBot
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18
votes
4
answers
2k
views
Does "taking the dual space" stabilize?
Every book which treats dual spaces of normend spaces states that $(c_0)' = \ell^1$ and $(\ell^1)' = \ell^\infty$ and some also describe $(\ell^\infty)'$.
However, is anything known about higher ...
- 16.2k
7
votes
1
answer
1k
views
If $H$ is a separable Hilbert space, is $L^2(H)$ separable?
Let $H$ be a separable Hilbert space, and let $\gamma$ be a Radon probability measure on $H$ with mean zero and covariance operator the identity $I$.
Is the Hilbert space $L^2(H,\gamma)$ separable?
CommunityBot
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7
votes
3
answers
1k
views
A Question concerning the Fourier Transform of $\mathbb{R}$
Consider the classical Schwartz space $\mathcal{S}(\mathbb{R})$ together with the Fourier transform $\mathcal{F} : \mathcal{S}(\mathbb{R}) \rightarrow \mathcal{S}( \mathbb{R})$.
Consider the subspace ...
- 61.9k
14
votes
5
answers
4k
views
Is there an extension of the Arzela-Ascoli theorem to spaces of discontinuous functions?
The Arzela-Ascoli function basically says that a set of real-valued continuous functions on a compact domain is precompact under the uniform norm if and only if the family is pointwise bounded and ...
- 279
1
vote
4
answers
614
views
Variants of point fixed theorem
Let $E$ be a dual Banach space and $C$ a nonempty convex weak* compact subset of $E$. Let $G$ be a group of weak* continuous linear isometries on $E$. Suppose that $g(C)\subset C$ for all $g\in G$.
...
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2
votes
1
answer
466
views
What is the regularity of the argument of a complex function?
Let $\psi=f+ig=\rho e^{i\theta}$ be a complex function on some open subset of $\mathbb{R}^n$, where $f,g,\rho$ and $\theta$ are real-valued. I happened to find that the identity of differentiation for ...
- 27.5k
0
votes
1
answer
1k
views
Precompact set in L2 space?
Let A be a bounded interval in R. Suppose we have a collection of functions, such that
Each function is $\in$ $L^r(A)$, where r is any number $\in$ $[1, \infty]$,
The fractional derivative of order ...
- 52.3k
5
votes
1
answer
1k
views
Lipschitz properties of minima/minimizers of convex functions of two variables
Suppose I have a function $f(x,y)$ from $\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ that is convex in both $x$ and $y$. Set
$g(y) = \min_{x} f(x,y)$
What I would like is for $g(y)$ to be ...
- 1,839
1
vote
0
answers
195
views
Lower semicontinuity of Bregman distances/divergences
For a Banach space $X$ and a convex functional $J:X \to [0,\infty]$ (i.e. with values in the extended reals), consider the associated Bregman distance: For $x,y\in X$ and $\xi\in\partial J(y)$:
\begin{...
- 12.7k
2
votes
1
answer
1k
views
Range of the Radon Transform
Let us consider the Radon transform in two dimensions:
$$\tag{1}Rf(r,\theta):=\int\limits_{-\infty}^{\infty} f(r\cos\theta-t\sin\theta,r\sin\theta+t\cos\theta) dt,$$
where $r\in\mathbb{R}$ and $0\...
- 931
-2
votes
1
answer
665
views
weak convergence
I know the following result is true in the case of strong convergence. But I don't know whether it is true in the case of weak convergence also.
Let $p>1$. Suppose that each $x_n$ is a non negative ...
- 779
29
votes
1
answer
4k
views
Furstenberg's Conjecture on 2-3-invariant continuous probability measures on the circle
Hillel Furstenberg conjectured that the only $2$-$3$-invariant probability measure on the circle without atoms is the Lebesgue measure. More precisely:
Question: (Furstenberg) Let $\mu$ be a ...
- 929
0
votes
1
answer
319
views
Hilbert space automorphisms realized as induced by transformations of some base-spaces
Following question may be soft. Fix abstract hilbert space H and consider any automorphism A in banach-spaces sence (i.e. no conditions on metric). Call A is realizable if exist measure space $(X,\mu)$...
- 22.3k
32
votes
2
answers
4k
views
Are there non-reflexive vector spaces isomorphic to their bi-dual?
Let $V$ be an infinite dimensional topological vector space and consider the natural application $\iota\colon V\to V^{**}$. The space $V$ is said to be reflexive if $\iota$ is an isomorphism.
Are ...
- 66.3k
0
votes
1
answer
296
views
Continuity of cylindrical functions.
Let $C_c^\infty(\mathbb R^n)$ be the functions from $\mathbb R^n$ to $\mathbb R$ with compact support, further let $X$ be a separable Hilbert space with a fixed orthonormal basis $(e_n)_n$. Define the ...
CommunityBot
- 1
8
votes
1
answer
431
views
Injectivity for bimodules and Hochschild cohomology
Let $A$ be a Banach algebra and let $X$ be an $A$-bimodule. Is there a notion of (relative) injectivity for $X$ which would imply that $\mathcal{H}^n(A,X)$ vanishes for all $n\ge 1$? Here $\mathcal{H}^...
CommunityBot
- 1
4
votes
2
answers
2k
views
Inclusions of $C^{k,\alpha}$ spaces
When is $C^{k,\alpha}(\bar{\Omega})$ a
subset of
$C^{k',\alpha'}(\bar{\Omega})$?
Gilbarg and Trudinger says that "for the domains of interest in this work the inclusion will hold whenever $k + \...
- 153
2
votes
2
answers
1k
views
Are coordinate functions on topological vector spaces always continuous?
Let $V$ be a Hausdorff locally convex topological vector space over the field $\mathbb{K}$.
Let $B$ be a subset of $V$ such that
$\;$ for all functions $c : B\to \mathbb{K}$, if $\displaystyle\sum_{...
- 31.5k
6
votes
0
answers
299
views
Spectrum of an operator arising in a dynamical problem
(Question edited according to Denis Serre comment).
While studying the action of dilating map of the circle on probability measures, I ran across the following operator:
$$\mathcal{K}^* : L^2_0(\mu)\...
- 14.4k
4
votes
1
answer
645
views
Factorization in the Wiener algebra on the unit disc.
Consider the Banach algebra $W^+=\ell^1(\mathbb{Z}^+)$, viewed upon as the analytic functions $f$ on the unit disc $\mathbb{D}$ such that $$\|f\|=\sum_{k\ge0}|a_k|<\infty$$ where
$$f(z)=\sum a_kz^k$...
- 36
7
votes
3
answers
4k
views
infinitely many linear equations in infinitely many variables
Let $(a_{mn})_{m,n\in\mathbb{N}}$ and $(b_m)$ be sequences of complex numbers.We say that $(a_{mn})$ and $(b_m)$ constitute an infinite system of linear equations in infinitely many variables if we ...
7
votes
3
answers
495
views
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, ...
CommunityBot
- 1
0
votes
1
answer
1k
views
Linear Mapping and integration
I have been reading the paper - "Introduction to Quantum Fisher Information".
In section 1.2 the author talks about the linear map $\mathbb{J}_D$, which he defines as follows:
Let $D \in M_n$ be a ...
- 2,066
6
votes
4
answers
7k
views
Why do we want to have orthogonal bases in decompositions?
In the decompositions I encountered so far, we all had orthogonal set of bases. For example in Singular Value Decomposition, we had orthogonal singular right and left vectors, in [discrete] cosine ...
- 12.7k
13
votes
6
answers
3k
views
Sets with equal positive measure in every interval
Hi,
I want to write a proof that relies on the fact that:
There are Borel Sets $A$ and $B$ contained in $\mathbb{R}$ such that
$A \cap B = \emptyset$ and $\lambda(A \cap (x,y)) = \lambda(B \cap (x,...
- 631
0
votes
1
answer
330
views
Convex sets and projections
Hello!
I recently started (it's purely self-education) reading a "Mathematical programming and optimizations" book, did a vast part of the exercises related to the theoretical part and at one moment ...
4
votes
2
answers
1k
views
Trace space and Neumann boundary condition
In which sense is it possible to solve $\Delta u=0$, $\partial_\nu u=\phi$, for $\int\phi=0$ on a closed domain, say a ball $B^3\subset\mathbb R^3$?
For example would a $\phi\in L^p(\partial B^3)$, $...
- 52.3k
7
votes
3
answers
1k
views
Non-Borel subspace of Banach space
Let $X$ be a separable Banach space, $M \subset X$ a linear subspace. Must $M$ be a Borel set in $X$?
I believe the answer is "no," since I have seen authors who are careful to talk about "Borel ...
CommunityBot
- 1
-1
votes
1
answer
2k
views
Absolute values and Frobenius norm [closed]
The Frobenius, or Hilbert-Schmidt, norm of an $n$ by $n$ matrix $A$ is defined as $\|A\|_2 = \sqrt{\sum_{i,j=1}^n |A_{ij}|^2}$. The absolute value of $A$ is the unique positive matrix $|A|$ satisfying ...
- 16
3
votes
1
answer
572
views
When is a finite matrix a "good" approximate representation of an operator?
I am interested in representing an arbitrary charge density (say, of atoms in a molecule) $\rho(r), \; r\in \mathbb{R}^3$ by a finite linear combination of basis functions
$\rho(r) = \sum_{i=1}^N q_i ...
CommunityBot
- 1
4
votes
1
answer
221
views
existence of charaterization of amenable groups by complementation?
Recall that we say that a closed space $F$ of a Banach space $E$ is complemented if there exists a contractive projection $P$ from $E$ onto $F$.
Do you know a charaterization of discrete amenable ...
- 18.7k
2
votes
4
answers
358
views
When do functions near F have zeros near a zero of F?
Consider a sequence of functions $F_n : \mathbb{R}^d \to \mathbb{R}^d$, a function $F: \mathbb{R}^d \to \mathbb{R}^d$, and an $\mathbf{x} \in \mathbb{R}^d$ so that $F(\mathbf{x}) = \mathbf{0}$. In ...
- 39.9k
6
votes
1
answer
453
views
The typical size of a random element in a Banach space
Let $X$ be a separable Banach space, and let $\mathbb P$ be a Radon probability measure on $X$ with zero mean and covariance operator $K : X^* \to X$. Let $x$ be an $X$-valued random variable with ...
- 11.4k
1
vote
1
answer
210
views
Is the metric obtained by altering the metric of a Hilbert space on a finite-dimensional subspace equivalent to the original one? [closed]
Suppose a Hilbert space W can be written as the direct sum (not necessarily orthogonal) of the closed subspaces H and V, where H is assumed to be of finite dimension. Define a new inner product via
||...
- 2,935
2
votes
2
answers
710
views
Are there good inequalities on the norm?
It's well known that in a Hilbert space, good inequalities exist concerning the norm due to the existence of inner product.Now let X be a general Banach algebra, are there good inequalities concerning ...
- 9,366
8
votes
2
answers
8k
views
Version of the Poincaré Inequality
Let $\Omega\subset \mathbb{R}^n$ open and bounded. The Poincaré inequality
$$\|u\|_p \le C \|\nabla u\|_p$$
($\|\cdot\|_p$ denotes the usual $L^p(\Omega)$-norm; the Lebesgue measure shall be used here)...
- 29.8k
0
votes
2
answers
377
views
"Frobenius-finite" linear operators on a Hilbert Space
Let $H = L_2(S)$ be the complex Hilbert space over $S$ with the counting measure. (There might be another term for this concept, but) I define a continuous linear operator $L$ on $H$ with matrix ...
- 15.3k
3
votes
2
answers
3k
views
Eigenvalues convolution-type operator
Let $J_1$ be the Bessel function of the first kind and let $H_1(x) = \frac{J_1(|x|)}{|x|}$ for $n = 1$. Define the operator $Tf(x) = (f * H_1)(x)$ from $L^2$ to $L^2$.
Since the $H_1$-function is the ...
- 1,981
1
vote
1
answer
506
views
Bessel sequence, uniformly minimal, separated
Is every unit norm Bessel sequence in a Hilbert space a finite union of separated ones? Is every unit norm separated sequence a finite union of uniformly minimal (minimal with uniformly bounded ...
- 31.5k
0
votes
1
answer
4k
views
2
votes
1
answer
1k
views
Green's function for wave equations in R² or R³
Hello,
For almost one year, I am searching for the Green's function for wave equation in R² or R³ with some boundary conditions. As far as I know, when the boundaries permit the method of images, we ...
- 52.3k
0
votes
3
answers
1k
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Sobolev norm and Beppo-Levi norm
I've asked this question on math.stackexchange.com but I'm not satisfied by the answers I got, so I've decided to ask here instead. As always I apologize if my notation is not precise enough. I am a ...
CommunityBot
- 1
1
vote
1
answer
706
views
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