Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [fa.functional-analysis]

Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

4
votes
1answer
177 views

Bounded growth of functions vs bounded growth of functions on countable sets

I am wondering if the boundedness of growth can be characterized by sequences. I am not sure if I use the term "growth" correctly, or use the correct tags for this question. Here is what I mean. Let $...
4
votes
1answer
83 views

Non-point spectrum for diagonalisable self-adjoint unbounded operator

Given a (separable) Hilbert space H and an unbounded densely defined linear operator $T:{\cal D}(T) \to $H such that ${\cal D}$ is diagonalizable (it means $\exists$ an O.N.B. of H such that all basis ...
2
votes
1answer
68 views

Dirac operator on manifold with periodic end

Let $\tilde W$ be a spin closed oriented manifold, $Y$ is a codimension $1$ closed oriented submanifold of $\tilde W$, and denote the $W$ the cobordism from $Y$ to itself obtained from cutting $\...
16
votes
1answer
497 views

Extreme points of convex compact sets

Preparing to a lecture on Krein--Milman theorem I read in W. Rudin's Functional analysis textbook (1973) that it is unknown whether any convex compact set in any topological vector space has an ...
1
vote
1answer
82 views

Is continuity preserved under norm operations

Let $F$ be a continuously differenable function over $\mathbb{R}$. Let $\Omega$ be a bounded subset of $\mathbb{R}^2$. Assume that for every $w\in L^2(\Omega)$ then $v(x)=F(w(x))$, $x\in \Omega$, is ...
2
votes
1answer
60 views

Regarding norm attaining functions

Let $X$ and $Y$ be Banach spaces.Let $L(X,Y)$ denote the space of all bounded linear map from $X$ to $Y$. $T:X\longrightarrow Y$ is said to be norm attaining if there exists a $x\in S_X$(the closed ...
4
votes
1answer
77 views

Is convolution jointly continuous on $\mathcal{E}'$?

Let $\mathcal{E}'(\mathbb{R})$ be equipped with its usual strong topology (being the dual space of $\mathcal{E}(\mathbb{R})$). Is convolution jointly continuous on $\mathcal{E}'(\mathbb{R})$?
4
votes
1answer
183 views

integral kernel function for the SU(N) group

It is well know that the Haar probability measure for the $U(N)$ group, given by $$ \begin{align} dX_{U(N)} & = \frac{1}{N!(2\pi)^N} \begin{vmatrix} 1 & 1 ...
3
votes
0answers
130 views

Direct proof of Closed Graph Theorem (or Bounded Inverse Theorem) from Uniform Boundedness Principle

I'm looking for a direct proof of the Closed Graph Theorem (or Bounded Inverse Theorem) from the Uniform Boundedness Principle. But I can't find one in the literature. I'm hoping there's a nice ...
3
votes
1answer
138 views

Universal decay rate of the Fisher information along the heat flow

I'm looking for a reference for the following fact: In the torus $\mathbb T^d$ let me denote by $u_t=u(t,x)$ the (unique, distributional) solution of the heat equation $$ \partial_t u=\Delta u $$ ...
0
votes
1answer
117 views

Weak convergences in Bochner spaces

I'm having bit trouble in understanding weak convergences in Bochner space. I have following question for some general measurable space $\Omega$: Let $\{x_n\}$ be a bounded sequence in $L^2((0,T)\...
1
vote
0answers
68 views

relative compact on nonlinear term

On the paper: Decay of Solutions to Nonlinear Schrodinger Equations. Let $u$ be a solution of the equation $$Hu+|u|^2u=0,$$ where $H$ is a Schrodinger operator, i.e. $-\Delta+V$ and $V$ is a (...
19
votes
1answer
538 views

Diameter of a quotient of the infinite dimensional sphere

Suppose a group $\Gamma$ acts by isometries on the Hilbert space $\mathbb{H}^\infty$ and it fixes the origin. So $\Gamma$ acts on the unit sphere $\mathbb{S}^\infty$ as well. Assume that the action $...
0
votes
0answers
61 views

Support size of a zero divisor

Let $G$ and $\mathbb C[G]$ be a torsion free group and its group algebra. Is there a function $f:\mathbb N\rightarrow\mathbb R$, with $\lim_nf(n)=\infty$ such that if $0\neq\alpha,\beta\in\mathbb C[G]$...
5
votes
1answer
130 views

Hilbert representation of a bilinear form

Let $\sigma:\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}$ be a bilinear symmetric form which is non-degenerate in the sense that for every $0\neq u\in \mathbb{R}^n$ there is $v\in \mathbb{R}^n$ with $...
9
votes
2answers
207 views

Trace on $\mathcal{S}(\mathbb{R}^k) \mathbin{\hat{\otimes}_\pi} \mathcal{S}'(\mathbb{R}^k)$

I asked this question on Math StackExchange, but it did not receive an answer, despite my offering a bounty to attract attention. I am unsure whether it is appropriate for this venue, but I thought ...
4
votes
0answers
109 views

Spectral gap for the Brownian motion with drift on a compact manifold

Let $M$ be a compact Riemannian manifold without boundary, $X$ a smooth vector field on $M$. Consider the Brownian motion $t\mapsto B_t$ on $M$ with drift $X$, so that its generator is $L=\Delta +X$. ...
0
votes
1answer
50 views

Covergent net in $\mathcal{E}'(\mathbb{R})$ implies bounded?

Let $\mathcal{E}(\mathbb{R})$ be the space of all $C^\infty$ functions on $\mathbb{R}$ with its usual topology, and $\mathcal{E}'(\mathbb{R})$ be the dual space with the weak* topology. Let $(T_i)_{...
6
votes
1answer
228 views

Convolution with semigroup: does this belong to the Sobolev space $W^{1,1}$?

Let $X$ be a Banach space, $T(t)$ be a strongly continuous semigroup on $X$, and $f\in L^1(0,\tau;X)$. It has been implied that the integral $$v(t)=\int_0^t T(t-s)f(s)ds,\quad t\in [0,\tau]$$ is not ...
5
votes
1answer
128 views

Unbounded version of continuous functional calculus

For a normal operator $T$ on a Hilbert space ${\cal H}$, it is well known that for any continuous complex valued function $f$ on the spectrum of $T$, we have a well-defined operator $f(T) \in B({\cal ...
0
votes
0answers
29 views

Existence of solutions to NLS: Local existence and boundedness

I was wondering when the following argument is valid: Consider a nonlinear Schrödinger equation $$i \partial_t \varphi = -\Delta \varphi+ N(\varphi)$$ where $N$ is a nonlinearity. Often it is ...
0
votes
0answers
29 views

trace embeddings and Sobolev inequality

I am reading the paper https://arxiv.org/pdf/0905.1257.pdf. Lemma 2.4. Equation (2.9) states that; When $n=1$, for any $U \in H^{1}_{0, L} (\mathcal C);$ its trace $U(x, 0)$ is continuous embedded in ...
8
votes
2answers
187 views

On the Fourier-Laplace transform of compactly supported distributions

Let $\mathcal{E}'(\mathbb{R})$ be the space of all compactly supported distributions on $\mathbb{R}$. For $f\in \mathcal{E}'(\mathbb{R})$, let $\widehat{f}$ denote the entire extension of the ...
1
vote
1answer
89 views

Optimal estimate in trace norm

Let $x,y$ be vectors of some Hilbert space of unit length. Then we can consider the projection $P_x:=\langle \bullet, x \rangle x$ and similarly $P_y.$ Assume then that we know that $\left\lVert x-...
1
vote
0answers
38 views

Representability of smooth invertible Lipschitz functions by a finite composition of near-identity functions

Theorem 1 of this paper shows that For every positve integer $K$ and for every nonempty bounded domain $\mathcal X \subseteq \mathbb R^d$, the restriction on $\mathcal X$ of any function $h: \...
1
vote
0answers
52 views

Completed Tensorproduct

I am trying to understand the completed tensorproduct. This can be defined as follows: Given a topological ring $R$ and two linearly topologized rings $A$ and $B$ with fundamental systems of open ...
2
votes
0answers
38 views

On different norms of the interpolating operator

Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
4
votes
1answer
67 views

Find $p$ s.t. there is a sequence of nodes in $[0,1]$ s.t. sequence of interpolating polynomials of every continuous function converges in $p$-norm

Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
3
votes
1answer
63 views

Is the compact-open topology on the dual of a separable Frechet space sequential?

Let $X$ be a separable Frechet space (= Polish locally convex linear metric space) and $X'_c$ be the space of linear continuous functionals on $X$, endowed with the compact-open topology (= the ...
1
vote
1answer
113 views

The completeness of spaces of continuous functions with the compact-open topology

For a Tychonoff space $X$ let $C_k(X)$ denote the space of continuous real-valued functions on $X$, endowed with the compact-open topology. Problem. Is the space $C_k(X)$ Polish if it is Polishable ...
4
votes
1answer
169 views

Given any sequence of interpolating nodes, can we find a continuous function $f$ whose interpolating polynomials doesn't converge to $f$ point-wise

Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
2
votes
1answer
135 views

Chain rules for Dini Derivative

Could someone provides some references for the chain rule concerning Dini derivatives. For example, let $f(\cdot) \in \mathcal{C}^1\left( \mathbb{R} ; \mathbb{R}\right)$, and $g(\cdot) \in \mathcal{C}\...
1
vote
0answers
41 views

Limit contration rates and expansion rate solenoid map

Let M:=$S^{1}\times \mathcal{D}^1$ where $\mathcal{D}=\{v\in \mathcal{R}^2 | |v|<1\}$ carries the product distance and suppose $f:M\rightarrow M$,$(x,y,z)\rightarrow (\gamma x, \lambda y+v(x), \mu ...
4
votes
0answers
63 views

Superposition operator from Sobolev space to Lebesgue space

Let $\Omega$ be a bounded, connected set in $\mathbb{R}^2$ with smooth boundary. I am wondering under what conditions on the real function $f(x):\mathbb{R}\to \mathbb{R}$ the superposition operator $F(...
2
votes
0answers
44 views

What restrictions on the form of an integral equation have a unique solution f=0?

We're stuck on the following question in a problem relevant to a physics paper on AdS/CFT that we are working on. Given a Fredholm equation of second kind with the form $f$+$\int_D K f\,dx = 0$, where ...
1
vote
1answer
51 views

Reference Request: Differentiability of Moreau Envelope

I've recently come across many results discussing the differentiation of the Moreau envelope defined by $$ e(f)(x)\triangleq \min_{h \in H} \|h-x\|^2 + f(z) , $$ where $f$ is a convex functional on a ...
1
vote
0answers
122 views

Compact embedding result

Let $\tau$ and $\ell$ be positive numbers. We know that the space $H^2(0,\ell)\cap H^1_0(0,\ell)$ is compactly embedded into $L^6(0,\ell)$. Now, is the space $L^2(0,\tau;H^2(0,\ell)\cap H^1_0(0,\ell))$...
1
vote
0answers
125 views

Why is $H^{1/2}$ a Hilbert space?

Let $n\in\mathbb{N}$ and $\Omega \subseteq \mathbb{R}^n$ sufficiently smooth. Then we have the Hilbert space $H^1(\Omega)$ and the trace operator $\operatorname{tr}: H^1(\Omega) \to L^2(\partial \...
2
votes
1answer
81 views

Fredholmness of formal selfadjoint operator $AA^*$ and Fredholmenss of $A$

Let $X$ and $Y$ be Hilbert spaces with respective inner products $\langle , \rangle_{X,Y}$. Let $A:X \rightarrow Y$ be a bounded linear operator. Assume there is a non-degenerate sesquilinear product $...
3
votes
0answers
34 views

Multi-parameter stationary phase asymptotic expansion

I am looking for an asymptotic expansion of the oscillatory integral of the form $$\int_{\mathbb{R}^n}f(x)\exp(i(\lambda_1\phi_1(x)+\dots+\lambda_k\phi_k(x))dx,$$ as $\lambda_i\to \infty$ ...
1
vote
0answers
81 views

How do we know the mollification is in the Sobolev space?

The first theorem of section 5.3. in Evan's PDE discusses approximating a function in $W^{p, k}$ by it's mollifications. Suppose $k$ is a positive integer, $1\leq p <\infty$ and $U$ is an open ...
1
vote
0answers
46 views

Confusion over dentability and denting points

I'm trying to learn about dentable sets and denting points in Banach spaces, but I'm running into some trouble. Part of the issue is that I've attacked this topic in a really sloppy, disorganised ...
2
votes
0answers
61 views

Sobolev extension with boundary condition

Let $\Omega$ be a Lipschitz bounded domain of $\mathbb{R}^n$, divided in two Lipschitz subdomains $\Omega_1$ and $\Omega_2$ such that $\Omega_1 \cap \Omega_2 = \emptyset$. We define the following ...
0
votes
0answers
94 views

Is there a precise relationship between ``Geometric Functional Analysis" and high-dimensional probability/information theory?

The 2009 course on GFA by Roman Vershynin (https://www.math.uci.edu/~rvershyn/papers/GFA-book.pdf) introduced the subject with this line on the course page, "...
4
votes
0answers
123 views

Cyclic vectors for regular representations

I'm looking for references about the following aspect of cyclic vectors for regular representations. Let $K$ be a compact Lie group. Let $K$ act on $L^2(K)$ by the left regular representation. Then $...
1
vote
1answer
102 views

Properties of Cameron Martin Space

In the case that I'm working with a separable Hilbert space, $H$, on which I have a trace class operator, $K$, that's coming from a Gaussian (i.e., $K$ is self-adjoint, and for simplicity, has trivial ...
3
votes
1answer
109 views

Schrödinger operator with Coulomb potential

The free Laplacian $-\Delta$ has absolutely continuous spectrum $[0,\infty).$ The Coulomb Hamiltonian $H=-\Delta-\frac{1}{\vert x\vert}$ on $L^2(\mathbb R^3)$ has absolutely continuous spectrum $[0,\...
3
votes
0answers
46 views

A question about a theorem in 'Quantum dynamical semigroups generated by noncommutative unbounded elliptic operators'

I have asked this question on stack and someone advised me to ask it here. The link is https://math.stackexchange.com/questions/2900658/a-question-about-a-theorem-in-quantum-dynamical-semigroups-...
10
votes
1answer
573 views

Quantum functional analysis

Can one explain some philosophy behind "quantum functional analysis" (or "quantized functional analysis") which was initiated and developed by such researchers as: Ruan Z.-J., Pisier J., Effros E.G., ...
10
votes
1answer
255 views

Criterion for a Banach algebra to be finite dimensional

Let $A$ be a Banach algebra (say, complex and unital) and suppose that every (closed) commutative subalgebra of $A$ is finite dimensional. Question. Does it follow that $A$ is finite dimensional? ...