All Questions
10,241 questions
7
votes
1
answer
261
views
Comparison of several topologies for probability measures
Let $X$ be a compact metric space and denote $\mathcal M(X)$ the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\operatorname{supp} \mu$ for the support of $\mu$. As is well ...
- 243
7
votes
1
answer
504
views
Are separable F-spaces (completely metrizable topological vector space) homeomorphic to $l_2$?
An F-space is a completely metrizable topological vector space, i.e. the vector topology is induced by a complete metric. A Fréchet space is, by definition, a locally convex F-space.
It is known that ...
- 183
7
votes
2
answers
460
views
Gaussian Surface Area of Positive Semidefinite Cone
Let $\mathbb{R}^n$ be the Euclidean space and $A \subseteq \mathbb{R}^n$ be a sufficiently regular set, e.g., one that has smooth boundary or is convex. We define the $\epsilon$-neighbor of $A$ in the ...
- 1,127
7
votes
1
answer
603
views
Is the equicontinuous weak-star topology locally convex on the dual of an LF-space?
The Banach-Dieudonné theorem states that if $X$ is a metrizable locally convex Hausdorff space then the equicontinuous weak-* topology $ew^*$ on $X'$ coincides with the topology of precompact ...
- 1,773
7
votes
3
answers
385
views
On what kind of condition of a compact set $K$ in the plane, $C(K)$ has a generator?
Let $K\subset \Bbb{C}$ be a compact subset of the complex plane, and let $C(K)$ be the space of all complex continuous functions on $K$.
We say that $f\in C(K)$ is a generator of $C(K)$ when the set $...
- 185
7
votes
1
answer
2k
views
Topology in space of test functions $\mathcal{D}(\Omega)$ and space of distributions $\mathcal{D}'(\Omega)$
We can concluded that $\mathcal{D}(\Omega):=\bigcup_{K \in \mathcal{K}(\Omega)} \mathcal{D}_K(\Omega)$ (where $\mathcal{K}(\Omega)$ denotes the union of all compacts set content in a open subset $\...
- 589
7
votes
2
answers
469
views
Invertibility of group Laplacian in $\ell^1$
Let $G$ be a discrete group and let $S$ be a generating set for $G$; assume that $S$ is symmetric (i.e., $g\in S$ iff $g^{-1}\in S$). Let $L=L_S=\frac{1}{|S|}(\sum_{g\in S} g-1)$ be an element of the ...
- 231
7
votes
1
answer
594
views
James' theorem—going from the separable case to the general case
Consider the following famous theorem by Robert C. James (1964):
Let $X$ be a Banach space over $\mathbb R$ and $C$ a non-empty, bounded, weakly closed subset. Then, $C$ is weakly compact if and ...
- 405
7
votes
1
answer
776
views
Fréchet-Kolmogorov compactness Theorem for Lp spaces on manifolds
Suppose I have a family of functions $\mathcal{F} \subseteq L^2(\mathcal{M}, P)$ where $\mathcal{M}$ is a compact manifold, and $P$ is a probability distribution on $\mathcal{M}$. Is there an ...
- 143
7
votes
1
answer
319
views
Why aren't operator semigroups studied from a dynamical perspective?
Often times one talks about iterating a continuous map to get discrete topological dynamics, or having a 1-parameter family of continuous maps to get continuous topological dynamics.
When studying ...
- 277
7
votes
1
answer
440
views
series representation in injective tensor products
All books on tensor products of Banach spaces contain the well-known theorem of Grothendieck that every element of the completed projective tensor product
$X \tilde{\otimes}_ \pi Y$ has a ...
- 16.5k
7
votes
1
answer
736
views
Banach Algebra Counterexample
Can someone give me an example of a Banach Algebra which does not have an isometric representation in a Hilbert Space ?
(if possible, can you add a proof or a reference ? )
Thank you very much !
...
- 73
7
votes
1
answer
463
views
Optimality of p-Lebesgue Differentiation Theorem for Sobolev Functions
This is the third question in a series whose purpose has been to flesh out an example of the optimality of the p-Lebesgue differentiation theorem for Sobolev functions. This theorem says that for $f \...
- 882
7
votes
2
answers
484
views
Extension of weakly compact operators from $\ell_1$ into $c_0$
Is every weakly compact operator from $\ell_1$ into $c_0$ extendible
to any larger space? Equivalently, is every weakly compact operator from $\ell_1$ into $c_0$ extendible to $\ell_\infty$?
7
votes
2
answers
1k
views
A book on Banach Manifold for a Dynamicist
Hi all,
Could you give me a suggestion of suitable book about Banach Manifolds for someone that have background in functional analysis at the level of Conway's book and Do Carmo's book on Riemannian ...
user11178
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 ...
- 71
7
votes
1
answer
243
views
Isoperimetric inequality, but $L_p$ norm
I would like to consider the isoperimetric problem of $L_p$ norm:
Given a region in $\mathbb R^2$ such that the boundary is a curve $C(x,y)$, where $\int_{C}(|\mathrm dx|^p+|\mathrm dy|^p)^{1/p}$ is a ...
- 843
7
votes
1
answer
152
views
Higher (BV) regularity of solutions to Poisson equation with Radon measure right-hand side?
I am trying to understand higher regularity for solutions to Poisson's equation when the right-hand side is a Radon measure. In particular:
$$\begin{cases}
\Delta u = \mu \text{ in } \Omega\\
u = 0\...
- 221
7
votes
1
answer
200
views
Projective tensor product of injective operators
I've seen claims that it is known that for a pair of bounded injective linear operators $T\colon X\to Y, S\colon W\to V$, their tensor product $T\otimes S\colon X \otimes_\pi W\to Y \otimes_\pi V$ ...
- 11.3k
7
votes
1
answer
384
views
Bounding the decrease after applying a contraction operator $n$ vs $n+1$ times
Can we upper bound the convergence rate of
$$\max_{\textbf{v}: \left\Vert \textbf{v}\right\Vert_2=1} \left\{ \left\Vert \textbf{T}^n \textbf{v}\right\Vert^2_2 - \left\Vert \textbf{T}^{n+1} \textbf{v}\...
- 673
7
votes
1
answer
547
views
Explicit isomorphism between $L^2(\mathbb{R}^2)$ and $L^2(\mathbb{R})$?
As Hilbert spaces, $L^2(\mathbb{R}^2)$ and $L^2(\mathbb{R})$ are isomorphic. Of course the isomoprhism is vastly not unique. I wonder if there are any particularly nice explicit isomorphisms. E.g. I ...
- 679
7
votes
1
answer
266
views
Visualizing the wave operator in two dimensions
For $n\geq 1$, let $D_n$ be the Dirac operator on the spinor bundle on the $n$-dimensional sphere $S^n$. For example, $D_1$ acts on the trivial bundle $S^1\times\mathbb{C}\to S^1$, and can be ...
- 1,913
7
votes
1
answer
201
views
Unitary representation is strictly continuous
Let $G$ be a compact group and $u: G \to B(H)$ be a strongly continuous unitary representation on the Hilbert space $H$. Then is $u: G \to B(H)$ strictly continuous?
That is, give $B(H)$ the topology ...
user167952
7
votes
1
answer
218
views
A hyperplane separation question
I have a subset $K\subset X^\ast$ of the dual of a Banach space $X$. (In fact $X$ is $C^1(M)$ for some smooth compact manifold $M$.) I hope that there exists $x\in X$ such that every $k\in K$ ...
- 55.9k
7
votes
2
answers
249
views
Do the operators in $B(E,F)$ separate points on the projective tensor product $F' \mathop{\tilde\otimes_\pi} E$?
Let $E$ and $F$ be Banach spaces, and let $\mathfrak L_{co}(E,F)$ denote the space of bounded linear operators $E \to F$ equipped with the topology of uniform convergence on the absolutely convex ...
7
votes
1
answer
251
views
Is a sign-preserving operator on $L^2$ a multiplication?
Let $T:L^2(\mu)\to L^2(\mu)$ be a linear and continuous operator, where $L^2(\mu)$ is the (real) $L^2$-space to some $\sigma$-finite measure space $(\Omega,\Sigma,\mu)$.
$T$ is assumed to be sign-...
- 273
7
votes
1
answer
2k
views
Regularity of Fourier transforms of $L^p$ functions for $2<p\le\infty$
I was recently reading about the Mikhlin and Hörmander Multiplier Theorems, which give conditions for a measurable function $m:\mathbb R^d\to\mathbb C$ to be an $L^p$ multiplier, i.e. for there to ...
- 1,247
7
votes
3
answers
2k
views
Collections of examples and counterexamples in (real, complex, functional) analysis, ODEs and PDEs
What books collect examples and counterexamples (or also "solved exercises", for some suitable definition of "exercise") in
real analysis,
complex analysis,
functional analysis,
ODEs,
PDEs?
The ...
user60665
7
votes
1
answer
2k
views
Property between trace class and Hilbert-Schmidt
Consider the following condition on a bounded operator $T$ on a Hilbert space:
$\ \ \ \ \ $(A) there exists an orthonormal basis $(e_j)$ with $\sum_j\parallel Te_j\parallel<\infty$.
We have the ...
user1688
7
votes
1
answer
509
views
Davis, Figiel, Johnson and Pełczyński factorization through spaces with a bases
Davis, Figiel, Johnson and Pełczyński's Factorization Theorem states that each weakly compact operator $T:X \to Y$ between Banach spaces $X$ and $Y$ factors through a reflexive Banach space $Z$. In ...
- 1,073
7
votes
1
answer
719
views
Tightness and Functional Analysis
Let $(\Omega , \mathbb{P})$ be a probability space and $X$ be a real-valued random variable. Then we immediately have the push-forward measure $\mu$ on $\mathbb{R}$ and one can think of $\mu$ as an ...
- 2,283
7
votes
1
answer
343
views
Is every $C_0$ semigroup on a Hilbert space automatically a $C_0$ group on a larger space?
Let $\{T(t),t\ge 0\}$ be a $C_0$ semigroup on a Hilbert space $X$, does that exist a larger Hilbert space $Y$ such that $X\subset Y$, and $T(t)$ extend to a $C_0$ group $T'(t)$ (so $t<0$ make ...
- 879
7
votes
2
answers
2k
views
Questions on topologies on space of Radon measures
Consider the space $C_c(\mathbb{R})$ of continuous real-valued functions on $\mathbb{R}$ equipped with the inductive limit topology by $C_c(\mathbb{R}) = \bigcup_{n \in \mathbb{N}} C_c(\mathbb{R}, K_n)...
- 1,773
7
votes
3
answers
2k
views
Derivatives of radial functions can be bounded by derivatives in terms of radial distance?
Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$,
and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$.
Prove or disprove the following.
Given any ...
- 398
7
votes
2
answers
641
views
Decay of solutions to Schrodinger equation with local minimum in potential
Consider the one-dimensional Schrodinger operator on the real line $\mathbb{R}$ given by
$$ L = - \partial_x^2 + V $$
where $V$ is a potential with the following properties:
$V$ is non-negative, and ...
- 39.1k
7
votes
1
answer
789
views
Subadditivity of the square root for matrices
For positive numbers $a$ and $b$ we have the inequality $\sqrt{a+b} \leqslant \sqrt{a} + \sqrt{b}$. Is it true that the same holds if we take $a$ and $b$ to be positive semidefinite matrices?
If not, ...
- 5,005
7
votes
2
answers
2k
views
Triangle inequality for $L^1$-norm with respect to a state
It is well-known that the naive construction of non-commutative $L^p$-spaces is performed only in tracial case. I would like to know if it is really a necessity.
To wit, let $\varphi$ be a normal ...
- 5,005
7
votes
2
answers
469
views
Eigenstates of Fourier transformation
Let $\gamma$ be defined on $\mathbb R^n$ by $\gamma (x)=e^{-π x^2}$. With $\mathcal F$ standing for the Fourier transformation defined on the Schwartz space by
$$
(\mathcal F u)(\xi)=\int e^{-2iπ x\...
- 16.2k
7
votes
1
answer
411
views
Banach spaces with no reflexive complemented subspaces
If $X$ is a Banach space with the Dunford Pettis Property (DPP), then no infinite reflexive subspace can be complemented. Suppose now that the Banach space has the property, that no infinite reflexive ...
- 149
7
votes
2
answers
2k
views
What functions can be obtained as a convolution of a Schwartz function and a tempered distribution?
Let $\mathcal S (\mathbb R)$ denote the space of Schwartz functions on $\mathbb R$ and $\mathcal S^* (\mathbb R)$ denote the dual space of Schwartz (a.k.a tempered) distributions.
We consider $\...
- 2,649
7
votes
1
answer
1k
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Exactness of completed tensor product of nuclear spaces
Let $0 \to V \to W \to L \to 0$ be a strict short exact sequence
of (complete) nuclear spaces, i.e. it is a short exact sequence of
(complete) nuclear spaces, all the maps are continuous, the map $...
- 2,649
7
votes
1
answer
2k
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On the Paley-Wiener theorem
Is there any even Schwartz function whose restriction to $[0,\infty)$ is monotone and whose Fourier transform is compactly supported? In other words, is there any entire analytic function satisfying ...
- 329
7
votes
1
answer
773
views
Equivalent metrics on Fréchet spaces and Lipschitz maps
Lipschitz maps are defined over metric space as maps $f:(X,d_X) \to (Y,d_Y)$ such that
$$ d\left( f(x),f(x^\prime) \right)_Y \le k d(x,x^\prime)_X \ \forall x,x^\prime \in X, $$
where $k$ is a ...
- 399
7
votes
1
answer
823
views
On a decomposition of L^1(G)
[EDITED by Y. Choi - I have attempted to paraphrase the original question into something a bit terser and more precise; if this is not what the original poster intended, they should make corrections ...
- 643
7
votes
1
answer
737
views
Question about projections on a Hilbert space
Sorry for the vague title, I can't think of a better one that isn't overly long.
Suppose that $S$ is a commuting set of projection operators on a Hilbert space. I'll introduce the following notation: ...
- 391
7
votes
2
answers
2k
views
What is the "Krein-Milman theorem for cones"?
Update: The question is completely answered. I had overlooked a reduction to the self-adjoint case, and the latter can be proved using a Hahn-Banach separation theorem. Thanks to Matthew Daws for ...
- 7,329
7
votes
2
answers
684
views
Yet more on distortion
I would like to elaborate a little bit on my previous question which can be found
here.
Firstly, let me recall that a separable Banach space $(X, \| \cdot \|)$ is said to be
arbitrarily distortable ...
- 576
7
votes
1
answer
253
views
Does a Banach algebra version of "the sum of a closed subspace and a finite dimensional subspace is always closed" exist?
In the setting of Banach spaces, it is well know that if $M$ is a closed subspace of a Banach space $X$ and $F$ is a finite dimensional subspace of $X$, then $M+F$ is closed.
Does a Banach algebra ...
- 515
7
votes
1
answer
184
views
Functional calculus on the Schwartz space instead of $L^2$?
As far as I know, functional calculus is typically carried out on Hilbert spaces with (possibly unbounded) self-adjoint operators.
However, I wonder if there is a way to do it on the space of test ...
- 3,477
7
votes
1
answer
185
views
Question on ODE involving mollifiers from Taylor's book on PDEs
In Taylor's third book on PDEs chapter 16, the author discusses quasilinear symmetric hyperbolic systems of the form
$$\partial_{t}u=A^{k}(t,x,u)\partial_{k}u+g(t,x,u)$$
with some initial condition $u(...
- 1,171