All Questions
10,049 questions
1
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0
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466
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Bounding point-wise maximum of the absolute difference of two convex functions
Let $\Delta: R \times R \rightarrow R_{+}$ be a positive and convex function (convex in, say, both the arguments) called the loss function.
Let $x \in R^d$. Moreover, let $H_1,...,H_r$ be sets of ...
- 11
8
votes
1
answer
1k
views
Applications of the Theorem of Gelfand-Naimark
Hi,
I am interested in the correspondence of algebraic results about C(X) (the space of continuous functions $X\to {\mathbb C}$(complex numbers) or $X\to {\mathbb R}$(real numbers) and topological ...
- 47.5k
2
votes
0
answers
200
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Fredholmness and invertibility in a C* algebra generated convolution-type operators
Let $PC$ be the algebra of complex-valued, piecewise-continuous functions from $[-\infty,+\infty]$, $SO$ be the algebra of bounded, continuous, complex-valued functions on $\mathbb R$ which are slowly ...
- 21
5
votes
3
answers
843
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Topology on the set of linear subspaces
Hello,
let $X$ be a seperable Hilbert space. Let $(e_i)_i$ be a Hilbert basis, and for each index let $E_i = \langle e_1,\dots,e_i \rangle \subset X$ the span of the first $i$ basis vectors. For any $...
CommunityBot
- 1
1
vote
0
answers
460
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Topology for test functions [closed]
One naive way to define a topology on test functions ${\mathcal D}(\Omega)$ would be to exhaust $\Omega$ by compacts $(K_n)$ and to take the metric induced by the semi-norm system
$$
{\| f \|} _ {n} :=...
- 45.7k
2
votes
1
answer
373
views
Strong measurability reference
I'm reading a book on Lyapunov Exponents by Lian and Lu in which they refer to strong measurability of operator-valued maps. They define this by saying an operator valued map $T:\Omega\to L(X,X)$ is ...
- 41.1k
11
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0
answers
309
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Combinatorial Hilbert spaces
Any closed subspace $V\subset {\ell}^2(\omega)$ has associated to it a subset ${\cal S}_V$ of ${\cal P}(\omega)$, call it a combinatorial Hilbert space, namely the set of all supports of all vectors ...
- 17.6k
12
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1
answer
859
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Who first found this characterization of Lebesgue integration?
Write $L^1$ for the Banach space $L^1([0, 1])$. Given $f \in L^1$, define $f_1, f_2 \in L^1$ by
$$
f_1(x) = f(x/2),
\qquad
f_2(x) = f((x + 1)/2).
$$
Let $I = \int_0^1$. Then $I$ is the unique ...
- 53.3k
2
votes
2
answers
867
views
Decomposition of an abelian von Neumann algebra
Hi, I came across the statement below and I couldn't figure out why it is true. I was hoping someone could explain it or give me a good reference. Thank you in advance.
"Let $\pi$ be a non-degenerate ...
CommunityBot
- 1
1
vote
1
answer
304
views
How do maximum norms relatively change in Euclidean translations
Let $Q$ be the cube $[-1,1]^{3}$ and $\pi$ be a plane in $\mathbb{R}^{3}$
that contains the origin but doesn't contain any vertex of $Q$. Suppose that $A$ is an invertible
linear transformation from $\...
- 11
3
votes
1
answer
1k
views
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. ...
CommunityBot
- 1
8
votes
3
answers
2k
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Definition of a von Neumann algebra
Is there a way to equip every C*-algebra A with a functorial topology such that
the canonical map A→A** is an isomorphism if and only if A is a von Neumann algebra?
Here A** denotes the dual of A* in ...
- 37.8k
11
votes
1
answer
806
views
Algebraicity of Eigenvectors in a Hilbert space
Let $(e_j)_{j\in\mathbb N}$ be an orthonormal basis of a Hilbert space $V$. Let $T:V\to V$ be continuous, selfadjoint linear operator.
Assume that for all $i,j\in\mathbb N$ the number $\langle Te_i,...
- 32.4k
7
votes
1
answer
2k
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A good reference for the wave front set
Hello,
I am wondering whether anyone know some good references for the theory of wave front set, microlocal analysis? I have some basic knowledge of distribution theory at the level of the Rudin's ...
- 5,293
14
votes
2
answers
926
views
"Explicit" embedding of $\ell^1$ as a closed subalgebra of a direct sum of matrix algebras
For sake of brevity let $A$ denote the Banach algebra formed by equipping $\ell^1({\mathbb N})$ with pointwise multiplication. This algebra is clearly not isomorphic as a Banach algebra to any uniform ...
CommunityBot
- 1
6
votes
1
answer
727
views
Is this method of "fractional sums" using a Fourier series viable?
Hi.
I have this idea about developing what I call a "continuum sum", that is, a method to "add up a non-integer number of terms", i.e. to see if there is a "natural" way to assign a meaning to the ...
- 1,687
7
votes
0
answers
273
views
Hilbert space from the Tate pairing
Fix an elliptic curve $E$ over ${\Bbb Q}$ (or if you prefer, something more general over something more general). For each extension $F$ of ${\Bbb Q}$, the Néron-Tate height pairing gives
an inner ...
- 17.6k
4
votes
1
answer
720
views
Are coordinate functionals on complete vector spaces always continuous?
(I'm just adding the completeness condition to $V$ from this 2 month old question of mine, because I realized it's relevant to whether Bill Johnson's answer to this 4 month old question of mine ...
CommunityBot
- 1
8
votes
1
answer
678
views
Spectral theory of pseudo-differential operators
Consider a finite rank complex bundle $E$ over $S^1$ with connection $\nabla$. Let $Q_0, Q_1 \in C^\infty(S^1, E)$ be pseudo-differential operators. $Q_0$ is defined by the symbol $\sigma_0(x, \xi) =...
- 579
3
votes
2
answers
413
views
Associativity of polar decomposition
By polar decomposition, every continuous linear function $f \colon H \to K$ between Hilbert spaces can be written uniquely as $f = \widehat{f} \circ |f|$ for a positive operator $|f| \colon H \to H$ ...
- 3,937
27
votes
3
answers
5k
views
Weak and Strong Integration of vector-valued functions
This is probably an elementary question, but outside my area of expertise, and I was unable to find any suitable reference:
Suppose $f:X\to E$ is a continuous function from a compact spaces (endowed ...
- 3,183
3
votes
1
answer
842
views
An elementary introduction of Colombeau's generalized function theory
Hello, I am wondering whether anyone know an elementary reference for Colombeau's theory on the multiplication of distributions? I encountered the problem of the square of Delta function. I need a ...
- 5,984
0
votes
1
answer
222
views
Bounding near the boundary for a Sobolev function.
Let $f: \Omega\rightarrow \mathbb{R}$ where $\Omega\subset\mathbb{R}^d$ is bounded with lipschitz smooth boundary. Further suppose that $f\in\mathcal{H}^{\tau}(\Omega)$, $\tau>\frac{d}{2}$ (i.e. $f$...
CommunityBot
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0
votes
1
answer
234
views
A property of "Schwartz" quadratic forms
Consider $K(x, y)$, $f(x)$ Schwartz functions and $g(y)$ a tempered distribution. Suppose $$K(x, y) = K(y, x)$$ Define
$$h(t) = \int f(x - t) K(x, y) g(y - t) dx dy$$
It appears to me $h(t)$ is a ...
- 1,368
10
votes
2
answers
960
views
Stone-Weierstrass for cones
A version of the Stone-Weierstrass Theorem asserts: If A is a linear subspace of C(K), the set of continuous functions on a compact space, and if A is a subalgebra that contains the constant functions ...
- 61.9k
1
vote
1
answer
491
views
Bounding a smooth function near the boundary
Let $f: \Omega\rightarrow \mathbb{R}$ where $\Omega\subset\mathbb{R}^d$ is bounded with lipschitz smooth boundary. Further suppose that $f\in\mathcal{H}^{\tau}(\Omega)$, $\tau>0$ and that $f$ ...
CommunityBot
- 1
4
votes
1
answer
2k
views
Characterizations of a linear subspace associated with Fourier series
Let $c_0$ be the Banach space of doubly infinite sequences $$\lbrace
a_n: -\infty\lt n\lt \infty, \lim_{|n|\to \infty} a_n=0 \rbrace.$$ Let $T$ be the space of $2\pi$ periodic functions integrable ...
- 744
6
votes
3
answers
3k
views
Can the adjoint of the exterior derivative in semi-Riemannian geometry be defined without the Hodge * operator?
The adjoint of the exterior derivarive is defined by
$\delta:=(-1)^k\ast^{-1}d\ast$,
but I need a way which avoids the Hodge $\ast$ operator.
Is there another definition?
For example, for ...
CommunityBot
- 1
5
votes
0
answers
616
views
Lebesgue measure on Frechet space?
It is well known that there are no Lebesgue measures on infinite-dimensional Banach spaces (see e.g. http://en.wikipedia.org/wiki/There_is_no_infinite-dimensional_Lebesgue_measure). However, I couldn'...
- 1,368
9
votes
2
answers
2k
views
Nice Classes of Non-Closable Operators
The only thing I know about non-closable operators can be summarised as "they exist, but they're nasty, so let's not talk about them!" This seems to be the case with everyone else I've talked to. I'd ...
- 511
2
votes
4
answers
1k
views
An inequality question
Let $M$ be a $3\times2$ matrix. Is it true that for any $x\in\mathbb{R}^{2}$
with $\left\Vert x\right\Vert _{3}=1$ there is some subspace $V$
with dimension $2$ of $\mathbb{R}^{3}$, such that $\left\...
- 45.7k
3
votes
0
answers
130
views
Positive block matrices over tensor algebras
Let $A$ be a unital C*-algebra. A positive block matrix in $M_2(A)$ must have the form
$$ \begin{pmatrix} a & a^{1/2} x b^{1/2} \\ b^{1/2} x^* a^{1/2} & b \end{pmatrix}, $$
where $a,b$ are ...
- 18.7k
5
votes
1
answer
781
views
Does a log-concave function on a convex set extend continuously to the boundary?
Let $U$ be an open convex set in a locally convex space $X$, and let $f : U \to [0,1]$ be a log-concave function on $U$ (i.e., bounded and real-valued). Under what conditions does $f$ have a ...
- 5,743
1
vote
1
answer
663
views
What are conditions to make f(x) defined by f(x)=f(x-1)*x + 1/e unique(for instance convex)?
[Background:]
Looking at the powerseries for the gamma-function
$ \Gamma(1+x) = 1 + a_1 x + a_2 x^2 - a_3 * x^3 + ... $
then we can arrive at a decomposition
$ \Gamma(1+x) = r(x) + g(x) $
...
CommunityBot
- 1
1
vote
2
answers
733
views
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
2k
views
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
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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
- 1
18
votes
4
answers
2k
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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
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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
- 1
7
votes
3
answers
1k
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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
0
votes
2
answers
2k
views
Vector spaces of random variables having zero expectation
Edit: Robin's comments appear to have made the matter a lot clearer to me. I now suppose that the vector space of random variables with zero expectation are studied in the context of second order ...
- 661
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 $g\in G$.
...
- 744
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
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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