All Questions
10,240 questions
6
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A Poincaré-like inequality
Is it true that for some real $K>0$ and all real $u\in C_0^\infty((0,1))$ we have
$$\int_0^1 (u'(x)^2+u(x)^2)\,dx\,\int_0^1 u(x)^2\,dx
\le K\Big(\int_0^1 x\,u'(x)^2\,dx\Big)^2\text{ ?}$$
- 128k
6
votes
1
answer
500
views
A characterization of metric spaces, isometric to subspaces of Euclidean spaces
I am looking for the reference to the following (surely known) characterization of metric spaces that embed into $\mathbb R^n$:
Theorem. Let $n$ be positive integer number. A metric space $X$ is ...
- 42k
6
votes
1
answer
574
views
Integration in Banach algebra
Let $\mu$ be a Borel measure on the real line $\mathbb{R}$ taking values in a separable Banach algebra $A$. Assume that $\mu$ is such that the total variation measure $|\mu|$ is finite. Let $f$ be a ...
- 552
6
votes
2
answers
514
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Convergence criterion in the domain of an unbounded operator
Cross-post from math.sx.
My question is somewhat close to this one, but the counterexamples given there do not apply here.
Setup. Given a Hilbert space $\mathcal H$, a closed operator $A$ and a ...
- 245
6
votes
2
answers
436
views
What properties should $C(M,\mathbb{R})$ have when $M$ is a $n$-dimensional manifold?
Let $M$ be a n-dimensional manifold, $C(M,\mathbb{R})$ be the function space of continuous function from $M$ to $\mathbb{R}$. What kind of properties should $C(M,\mathbb{R})$ has, to reflect the ...
- 523
6
votes
3
answers
717
views
Function of moderate growth: history, motivation, and uses
I recently came across functions of moderate growth via Are functions of moderate growth a bornological space? and I was wondering, what are some concrete uses or applications of this space? Where ...
- 5,405
6
votes
1
answer
241
views
Self-adjointness and choosing appropriate function spaces
Consider the following operator on some (yet undecided) space $S$ of functions over $[0\:\:1]$
$$L(u)=\sin(x)u-x\dfrac{\partial u}{\partial x}$$
Now, its formal adjoint is
$L^*(v)=\sin(x)v+\dfrac{\...
- 318
6
votes
2
answers
548
views
When is it $C(X)$?
Suppose that $\tilde{X}$ is a compact space. If $C(\tilde{X})$ is isometrically isomorphic to the second dual of a Banach space, does there exist a locally compact space $X$ such that $C(\tilde{X})=...
- 306
6
votes
4
answers
1k
views
Reference for integral of functions taking values in a topological vector space.
(Note that I am interested in the Gelfand-Pettis integral specifically, as opposed to, for example, the Bochner integral.) I have tried Googling things like "integral topological vector space", "...
- 1,270
6
votes
2
answers
2k
views
Weak convergence + convergence of the norm implies strong convergence in Orlicz spaces
It is known [1, proposition 3.32] and a classical trick in PDEs that, in any uniformly convex Banach space $X$, weak convergence $x_n\rightharpoonup x$ together with convergence of the norm $\|x_n\|_X\...
- 5,380
6
votes
2
answers
644
views
Explicit form of this unitary transformation
Disclaimer: This question has its motivation from physics. It is probably not entirely rigorous at the moment. I just want to clarify some steps and try to make the arguments rigorous afterwards, if ...
- 3,535
6
votes
2
answers
424
views
Lipschitz mappings, covering dimension
Is there a compact metric space $X$ of covering dimension $2$ without a Lipschitz surjection on $[0,1]^2$?
For a space $X$ with Hausdorff dimension greater than $2$, we have a negative answer (see ...
- 97
6
votes
2
answers
326
views
Looking for references to study $U^p$ and $V^p$ spaces
I am studying some papers in the analysis of nonlinear PDEs and I am encountering the $U^p$ and $V^p$ spaces for the first time. Where can I find references more detailed than papers?
Edited
The ...
- 159
6
votes
1
answer
171
views
Some special sequence in $C(\mathbb{R})$
Let us consider $C(\mathbb{R})$, the space of continuous functions on the reals.
Q. Does there exist a sequence $\{f_n\}$ in $C(\mathbb{R})$ such that for every $f\in C(\mathbb{R})$ one may find a ...
- 4,058
6
votes
2
answers
539
views
Is there a reasonable notion of spectral theorem on a pre-Hilbert space?
I'm trying to understand how bad things could possibly get without Cauchy completeness as a criterion for Hilbert spaces in quantum mechanics. Obviously, doing calculus on a pre-Hilbert space would be ...
- 269
6
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1
answer
619
views
Whether Krein-Milman property implies Radon-Nikodym property
A Banach space is said to have Krein-Milman property (KMP in short) if every closed bounded convex set of it is a closed convex hull of its extreme points. Eg. Any reflexive space has KMP, $\ell_1$ ...
- 521
6
votes
2
answers
1k
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Properties of heat equation
** I simplified the question: **
On bounded domains, the maximum principle implies that the solution to the heat equation is (strictly) positive, if the initial and boundary data is positive.
I ...
user121558
6
votes
2
answers
353
views
Bounded deformation vs bounded variation
Let $BV(\mathbb R^n; \mathbb R^n)$ be the space of (vector-valued) functions of bounded variation and let $BD(\mathbb R^n;\mathbb R^n)$ the space of functions with bounded deformation. They are made ...
- 608
6
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1
answer
577
views
Who gave the generalized Stone-Weierstrass Theorem?
Let $X$ be a compact Hausdorff space and $\mathcal{A}$ be a closed self-adjoint subalgebra of $C(X)$ which contains the constants. Then $\mathcal{A}$ is the collection of continuous functions on $X$ ...
user14319
6
votes
1
answer
1k
views
Left invertible operators of $B(X,Y)$
Suppose that $X$ and $Y$ are Banach spaces. Is $\{f\in B(X,Y):f\ \text{has a left inverse}\}$ an open subset of $B(X,Y)$?
- 591
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2
answers
2k
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What is the translation in Fourier transform for a function to have exp. decay at $x\to -\infty$
It is known that smooth functions with exponential decay at $\pm\infty$ are functions whose Fourier transform have analytic continuation in some suited complex strip. I was wondering what happens if ...
- 319
6
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4
answers
1k
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Existence of dominating measure for weak*-compact set of measures
I have posted the following question also here a longer time ago, but due to no answers I thought it might fit better to MO.
Let $(\Omega,\mathcal F)$ be a measurable space and $\mathcal P$ a weak*-...
- 215
6
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1
answer
754
views
Banach Manifold
Let $M$ and $N$ be closed manifolds. Is it true that
$C^{k}(N,M)$, which is the space of functions $f: N\to M$ such that $f\in C^{k}$, is a $C^{\infty}$ Banach manifold? If so, can you help me to ...
- 225
6
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6
answers
1k
views
Proving continuity on spaces of distributions?
Let $\mathcal{D}'(\Omega)$ be the space of distributions on an open set $\Omega$, and $\mathcal{E}'(\Omega)$ the compactly supported ones.
When you have a linear operator $T:\mathcal{D}'(\Omega)\...
- 61
6
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2
answers
4k
views
Is there dual space of the distributions $\mathcal{D}'(R)$?
Dear MOs,
Let $\mathcal{D}(R):=C_c^\infty(R)$ be the smooth functions with compact support. Its dual space is the space $\mathcal{D}'(R)$ of distributions. This space $\mathcal{D}(R)$ has its weak *-...
- 1,649
6
votes
4
answers
8k
views
Characterization of the non-negative definite functions $f(x,y)$
The common definition of the non-negative definite functions is as follows:
Definition 1: A continuous complex-valued function $f(x)$ is called non-negative definite, if for any real numbers $x_1,\...
- 1,649
6
votes
2
answers
4k
views
Bounded and weakly bounded sets in top. vector spaces
Consider a locally convex topological vector space V over the complex numbers. Is it true that every weakly bounded subset of V is indeed bounded? If not, what additional requirements are needed for ...
- 61
6
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1
answer
288
views
Sigma-weakly dense *-subalgebra of von Neumann algebra has increasing net of positive elements convergent to the identity
Let $M$ be a von Neumann algebra and $A\subseteq M$ a $\sigma$-weakly dense $*$-subalgebra of $M$. Does there exist an increasing net $\{a_i\}_{i\in I}\subseteq A\cap M^+$ such that $a_i\to 1$ in the $...
- 175
6
votes
2
answers
463
views
Spectrum of operator involving ladder operators
The ladder operator in quantum mechanics are the operators
$$a^\dagger \ = \ \frac{1}{\sqrt{2}} \left(-\frac{d}{dq} + q\right)$$
and
$$a \ \ = \ \frac{1}{\sqrt{2}} \left(\ \ \ \!\frac{d}{dq} + q\...
6
votes
2
answers
348
views
Does there exist a framework for determining if a power series is "differentially algebraic"
It is a well studied problem to take a function $f$ expressed (usually expressed as a solution to a differential equation w/ some initial conditions) and ask if it has an "elementary closed form&...
- 2,689
6
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2
answers
333
views
Is there a way to reconstruct the convolution $(f * g)(x)$ of $f$ with a Gaussian $g$ from sampled values, $(f*g)(a), a \in A$?
Suppose that $f: \mathbb{R} \to \mathbb{C}$ is a function which has support in $[-1,1]$. Let $g = g_\sigma$ be a centered Gaussian with variance $\sigma^2$. Is there a way to reconstruct the ...
- 437
6
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1
answer
338
views
Atiyah-Singer for Riemannian and Kaehler manifolds
I am trying to understand the proof of the Atiyah--Singer index theorem, and would like to see how it works for two "simple" examples. Could somebody direct me to a proof for the special ...
- 203
6
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1
answer
214
views
Are lattice operations in a Lipschitz space sequentially continuous in the weak* topology?
This is a follow-up on this (answered) question on math.SE, but involves a different topology. I think this time it is more appropriate for MO. I will repeat the background from the question cited ...
- 437
6
votes
1
answer
271
views
Approximation property counterexamples? (Also: relation to tensor products)
I remember reading somewhere (but unfortunately, I've forgotten where it was) that the canonical map from the (completed) projective tensor product of two Banach spaces to the (completed) injective ...
- 508
6
votes
2
answers
486
views
Equivalence classes of norms on $R^n$ under symmetries
Let $G \leq {\bf GL}_n$ be a symmetry group on $\mathbb{R}^n$. For simplicity, we can consider the case $G = {\bf GL}_n$.
Define two norms $\|\cdot\|_1$ and $\| \cdot\|_2$ to be equivalent under $G$ ...
- 252
6
votes
2
answers
735
views
Tensor product space with projective norm is incomplete
Ryan says in his book "Introduction to Tensor Products of Banach Spaces"(pg. 17) that for Banach spaces $X$ and $Y$, $X\otimes Y$ equipped with projective norm is not complete unless $X$ and $Y$ are ...
- 163
6
votes
3
answers
266
views
Approximating dense subspaces of Fréchet spaces
If $H$, $H_0$ are two separable Hilbert spaces and $H$ is continuously and densly embedded in $H_0$, it is possible to construct a sequence of linear operators
$$ P_n : H_0 \to H $$
such that for all $...
- 570
6
votes
3
answers
601
views
Differential calculus of functions of self-adjoint operators
Let $H$ be a Hilbert space over $\mathbb{C}$. Fix a self-adjoint operator $A:D(A)\rightarrow H$ and a Borel function $f:\mathbb{R}\rightarrow\mathbb{C}$. The operator $f(A)$ is defined by the spectral ...
- 61
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2
answers
3k
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Closed convex bounded sets are weakly compact for which spaces?
It is known that for all reflexive Banach spaces, closed convex bounded sets are weakly compact (compact for the weak topology).
What is the general class of topological vector spaces for which this ...
- 549
6
votes
1
answer
277
views
Diagonalization of the matrix $(1/(i+j+\rm{const}))_{i,j}$
Consider the following infinite matrix: $A_{i,j}=\frac1{i+j+\gamma}$, $0\leq i,j<\infty$, $\gamma>0$ is a constant. Is it known how to diagonalize $A$, or, say, calculate $(I+tA)^{-1}$ for ...
- 109k
6
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1
answer
760
views
Example of an infinite dimensional reflexive Banach algebra
If a $C^\ast$-algebra is reflexive (as a Banach space) then it is finite dimensional. Can anyone provide (or give a reference to) a nice example of an infinite dimensional non-commutative Banach ...
- 777
6
votes
1
answer
428
views
Poincaré lemma in infinite dimensions
Hi everyone,
Is the Poincaré lemma true in infinite dimensions?
Here's a precise statement:
Let $X$ be a Banach (or maybe Hilbert) vector space, $U$ a simply connected open set in $X$. Is it true ...
- 1,347
6
votes
2
answers
1k
views
Commuting Linear Operators In Hilbert Spaces
Let $V$ be a finite dimensional vector space over the complex field $\mathbb C$. Let $L:V\rightarrow V$ be a linear operator. Using the matrix of $L$ and the Jordan canonical form it is easy to find ...
- 545
6
votes
2
answers
605
views
$\beta\mathbb{N}$ vs $\beta\mathbb{Z}$
Just started learning the Stone-Cech compactification of discrete groups this week. My motivation comes from a question on $\beta\mathbb{Z}$. Surprisingly, I realized there are muchhhh more literature ...
6
votes
2
answers
979
views
Literature on behaviour of eigenfunctions under multiplication?
Dear community,
I would be happy about any literature or comments on the behaviour of the pointwise product of eigenfunctions of a self-adjoint operator with discrete spectrum, acting on a separable ...
- 199
6
votes
2
answers
909
views
Do maps have flows?
In A New Kind of Science: Open Problems and Projects(pg. 36).
How can one extend recursive function definitions to continuous numbers? What is the continuous analog of the Ackermann function? The ...
user37691
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
6
votes
3
answers
282
views
Extreme points of the dual unit ball of a Banach algebra
Let $A$ be a unital Banach algebra. Let $f\in A^*$, $\Vert f\Vert=1$ satisfy that there exists a maximal left ideal $L\subset A$ such that $L\subseteq\ker{f}$.
Question: Is $f$ an extreme point of ...
- 2,605
6
votes
1
answer
528
views
A functional equation
I am working on some physics problem and got stuck with the following equation: Let $a$ be a very small positive number. Is there a bounded function $F$, $0 \leq F \leq 1$, such that for all $x \in \...
- 63
6
votes
1
answer
436
views
Diagonalizing the ‘restricted’ Hilbert transform on $L^2(0,1)$, $f(z_1) \mapsto \mathrm{p.v.} \int_0^1 \frac{i}{z_1-z_2}f(z_2) dz_2$
Consider the following operator on functions $\mathcal{T}: L^2(0,1) \to L^2(0,1)$ over the complex numbers.
\begin{equation}
(\mathcal{T} f)(z_1) = \mathrm{p.v.} \int_0^1 \frac{i}{z_1-z_2}f(z_2) dz_2
...
- 545