Questions tagged [fa.functional-analysis]

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

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2 votes
1 answer
69 views

Are projective tensor products left-exact if one considers only maps of norm at most 1?

Consider the category $\mathrm{Ban}$ of Banach spaces and bounded linear maps and the category $\mathrm{Ban}_1$ of Banach spaces and bounded linear maps of operator norm at most 1. Let $\otimes_\pi$ ...
0 votes
3 answers
179 views

A generalisation of Tchebychev inequality

Let $f,g \in C(\mathbb R)$ with $\exists M \in \mathbb R^*, \forall (x,y) \in \mathbb R^2, M\times (f(x)-f(y))(g(x)-g(y)) \geq 0$. Is it true that exists $ u$ any real function, and $a,b$ monotone ...
0 votes
0 answers
43 views

Unusual boundary value problem

In F.V. Atkinson's book, Discrete and continuous boundary value problems, (1964), there is the following BVP described in chapter 2.11: \begin{equation} \begin{cases} y'=-\frac{i}{\lambda}...
5 votes
0 answers
727 views

Is such a Banach space $X$ isometrically isomorphic to a Hilbert space or not?

Let $X$ be a real or complex Banach space. $X$ satisfies:There exists real or complex series $\{a_k\}_{k=1}^n,\{b_k\}_{k=1}^n$ (which satisfies that:$\begin{cases} a_k,b_k\in \mathbb{R}\ \forall k=1,\...
2 votes
1 answer
82 views

$L^2$ space of Hilbert-Schmidt operator valued functions

Let $\mathscr{S}$ denote the space of all Hilbert-Schmidt operators on $L^2(\mathbb R)$. Consider the Hilbert space $L^2(\mathbb R, \mathscr S)$ of square-integrable $\mathscr S$-valued function, that ...
1 vote
0 answers
23 views

Convergence of slice in an equivalent renorming

Let us consider $\ell_2$ space with $\Vert \cdot \Vert_2$ norm. Let us define a new norm equivalent to $\Vert \cdot \Vert_2$ norm as follows: $$ \Vert x \Vert_0 = \max \{ \Vert x \Vert_2, \sqrt{2} \...
7 votes
1 answer
216 views

Is the dual of a Fréchet space weakly* separable?

It is known that if $X$ is a separable Banach space with dual $X^\ast$, then $B_{X^\ast}$, the closed the unit ball in $X^\ast$, is compact and metrizable in the weak* or $\sigma(X^\ast, X)$-...
1 vote
1 answer
61 views

$L^p$ domination of mixed partial derivatives of the 3rd order by the unmixed ones?

Is it true that for each real $p>1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^3$ one has $$\|D_1D_2D_3u\|_p\le C_p(\|D_1^3u\|_p+\|...
2 votes
1 answer
197 views

Decompose a function into a bounded part and a Lipschitz part

Let $f: \mathbb R^d \to \mathbb R^d$ be a measurable function such that $$ \sup_{x,y \in \mathbb R^d} \frac{|f(x) - f(y)|}{\max \{1, |x-y| \}} < \infty. $$ Are there functions $g,h: \mathbb R^d \...
2 votes
2 answers
154 views

$L^p$ domination of mixed partial derivatives by the unmixed ones?

Is it true that for each real $p\ge1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^2$ one has $$\|D_1D_2u\|_p\le C_p(\|D_1^2u\|_p+\|D_2^...
5 votes
2 answers
668 views

How can I prove this special version of the Poincaré inequality?

I am reading a paper, Is $L^2$ Physics-Informed Loss Always Suitable for Training Physics-Informed Neural Network?, which uses an "application" of Poincaré's inequality. The form I know and ...
2 votes
1 answer
137 views

Are these two norms on localized versions of $L^p_q$ equivalent?

$\newcommand{\RR}{\mathbb R}\newcommand{\diff}{\, \mathrm d}$ We fix $T \in (0, \infty)$ and $p, q \in [1, \infty)$. Let $\mathbb T$ be the interval $[0, T]$. Let $E$ be the space of all real-valued ...
3 votes
1 answer
245 views

Schroedinger operator in 2 dimensions with singular potential

Consider the Schroedinger operator $$H = -\Delta + \frac{c}{\vert x \vert^2}$$ in two dimensions with $c >0$ This operator has a self-adjoint realization, since it is a positive symmetric operator ...
2 votes
0 answers
45 views

Constructing spaces of analytic sections. Silva spaces

This is a follow up on this question Representing geodesic compactifications of $S^1\times \Bbb R$ as analytic sections over base (analytic) foliations to add more details. In the linked question I ...
1 vote
0 answers
42 views

Fractional powers of Dirichlet-to-Neumann map to derive estimate for PDE

Assume $\Omega$ is an open, bounded subset of $\mathbb R^3$ with smooth boundary $\partial \Omega= \Gamma$. For $u \in H^{1/2}(\Gamma)$, let $U \in H^1(\Omega)$ denote the weak solution of the ...
0 votes
0 answers
176 views

Eigenvalue multiplicity of tensor product of positive operator with itself

Let $H$ be a separable complex Hilbert space and let $A\in B(H)$ be positive with $||A||=1$ and have eigenvalue 1 with multiplicity 1. Suppose $A=T^*T$ for some $T\in B(H)$. Denote the spectrum of $A$ ...
1 vote
0 answers
63 views

positive invertible maps which are not *-automorphisms

Let $A$ be a unital C*-algebra. Is there a unital positive self-map $F:A\to A$ which is invertible (i.e. injective and surjective) but not a $*$-automorphism? If yes, how does appear its Gelfand-...
1 vote
2 answers
2k views

Bounding the norm of the Laplacian of the gradient of a function having Lipschitz continuous Hessian

It seems that the following claim is true, but I did not manage to prove it neither to find a reference. Claim Let $f:\mathbb R^p\to\mathbb R$ be a three times differentiable function such that its ...
0 votes
1 answer
107 views

Singular integral bounded by Dirichlet form?

We define for some fixed $L$ $$\Omega:=\{(x_1,x_2) \in ([-L,L]^2 \times [-L,L]^2) \setminus \{x_1=x_2\}\},$$ in particular $x_1,x_2 \in \mathbb R^2.$ Let $f \in C_c^{\infty}(\Omega)$, then I am ...
1 vote
1 answer
89 views

Image of a complete topological group under open and surjective map is complete?

A uniform space $X$ is complete if every Cauchy filter in $X$ is convergent. Here we do not require $X$ to be Hausdorff. Question. Let $G$ be a complete topological group and let $H$ be a topological ...
0 votes
0 answers
55 views

asymptotic expansions for $C^{1+\epsilon}$operators

I want to understand the asymptotic expansion for $C^{1+\epsilon}$ operators. More precisely, if a complex operator $T(z)=\sum_{n \ge 1}T_n z^n$ is defined on a closed unit disk, and it is $C^{1+\...
2 votes
1 answer
121 views

Difference in essential spectrum between Schrodinger operators

I am considering two Schrodinger operators on $\mathbb{Z}^2$ and compare their essential spectrum. The operators are both of the form $H=A+V$ where $A$ is the adjacency operator on the $\mathbb{Z}^2$-...
4 votes
1 answer
655 views

Question/References on the Skorokhod M1 topology

Let $D(0,T)$ be the space of right continuous functions with left limits defined on $[0,T]$. Consider the Skorokhod M1 topology on $D(0,T)$, see e.g. S. Ledger, Skorokhod’s M1 topology for ...
2 votes
0 answers
52 views

Commutant of rank one operators over Hilbert module

Let $\mathcal{E}$ be a right Hilbert $C^*$-module over the $C^*$-algebra $A$. Consider for $\xi, \eta\in \mathcal{E}$ the rank-one operator $$\theta_{\xi, \eta}: \mathcal{E}\to \mathcal{E}: \zeta \...
-2 votes
0 answers
35 views

Convergence of function sequences derived from systems of linear equations in operations research

I am working on a problem in operations research that involves systems of linear equations with a specific structure, and I am seeking insights or references that might help in proving a pointwise ...
3 votes
1 answer
125 views

Rescaling Fourier coefficients of a continuous function by a bounded sequence

This question stems out of: which sequences $(a_n)_{n\in\mathbb{Z}}$ of complex numbers have the property that if there exists a continuous function $f$ on the circle with Fourier coefficients $b_n$, ...
6 votes
2 answers
225 views

If a Banach / Fréchet manifold $M$ happens to be a topological vector space, is $M$ just a Banach / Fréchet space?

In finite dimensions, if $M$ is a smooth manifold that happens to be a vector space, then it is indeed just the Euclidean space. I wonder if the same result holds valid in infinite dimensions. More ...
1 vote
1 answer
68 views

Convexity property of an equivalent norm on $\ell_2$

Let us consider the space $\ell_2$ with an equivalent norm defined by $$ \Vert x \Vert = \max \{ \Vert x^{'} \Vert_2, \Vert x^{''} \Vert_2 \}, $$ where $x^{'}=(0, x_2, x_3, \cdots)$, $x^{''} = (x_1, 0,...
4 votes
0 answers
55 views

Find reasonable definition for endpoint Lorentz function spaces $L^{\infty,q}$ via the idea from endpoint Triebel-Lizorkin ${\scr F}_{\infty,q}^s$

On a measure space $(X,\mu)$, for $0<p,q<\infty$ the Lorentz space $L^{p,q}(\mu)$ is defined by $$\|f\|_{L^{p,q}(\mu)}:=p^\frac1q\|t\mu(|f|>t)^\frac1p\|_{L^q(\mathbb R_+,\frac{dt}t)}=p^\...
4 votes
1 answer
153 views

Reference request: Gaussian measures on duals of nuclear spaces

I am interested in constructive quantum field theory where Gaussian measures on duals of nuclear spaces (specifically, the space of tempered distribution $\mathcal{S}'(\mathbb{R}^n)$) play a key role. ...
2 votes
1 answer
461 views

Shift-invariant spaces

We can define a shift-invariant space as $$V_{\varphi}(\mathbb{Z}):=\left\{\sum_{k\in\mathbb{Z}}c_k\varphi({\cdot}-k):(c_k)\in \ell_2\right\},$$ where convergence of the series is taken to be in $L^2(\...
1 vote
0 answers
54 views

Is the adjoint action of $\mathrm{SU}(2)$ on the Schwartz space proper and free?

$\DeclareMathOperator\SU{SU}$Let $t_1, t_2, t_3$ be generators of the Lie algebra $\mathfrak{su}(2)$. Let us consider a Schwartz space $\mathcal{S}$ defined as \begin{equation} \mathcal{S}:= \Bigl\{ \...
10 votes
1 answer
219 views

Which closed subsets $Y$ of a compact space $X$ admit a linear extensor $C(Y)\to C(X)$?

In the following $X$ is a Hausdorff compact topological space. Let $Y$ be a closed subset of $X$. The restriction operator $R_Y:C(X)\to C(Y)$ is surjective (Tietze), so it admits a continuous right ...
65 votes
14 answers
5k views

Notions of convergence not corresponding to topologies

This question concerns the ramifications of the following interesting problem that appeared on Ed Nelson's final exam on Functional Analysis some years ago: Exam question: Is there a metric on the ...
2 votes
1 answer
135 views

Invertibility of message passing with invertible parametrization

Consider the message passing framework defined by, $$f(\boldsymbol{x}_i)= \boldsymbol{x}_i + \sum_{j \neq i} (\boldsymbol{x}_i -\boldsymbol{x}_j) g(\|\boldsymbol{x}_i -\boldsymbol{x}_j\|^2),$$ for $i=...
1 vote
1 answer
92 views

Maximal element w.r.t. abolute continuity of measures

Suppose that $\mu$ is a $\sigma$-finite measure on $\mathcal{X}\equiv\bigotimes_{i=1}^n\mathcal{X}_i$. Let $\Pi$ denote the set of all $\sigma$-finite product measures on $\mathcal{X}$. Define $$ \...
5 votes
1 answer
162 views

Can the Wiener Chaos expansion converge in $L^p$?

For a function $F$ satisfying $|F(x)|\le C(1+|x|)^M$ for some $C,M>0$, and a mollifier $\rho$, we define $F_{\epsilon}=F\ast\rho_{\epsilon}$, where $\rho_{\epsilon}(x)=\epsilon^{-1}\rho(\frac{x}{\...
3 votes
1 answer
76 views

Can gradient zero implies that a function is constant with Hörmander vector fields

Let $X=(X_1,\cdots,X_m)$ be a system of Hörmander vector fields defined on $\mathbb{R}^n$. The Sobolev space $W_{X}^{1,p}(\Omega)$ is defined by $$W_{X}^{1,p}(\Omega):=\{u\in L^p(\Omega)|X_iu\in L^p(\...
6 votes
4 answers
7k 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,\...
57 votes
23 answers
101k views

A good book of functional analysis [closed]

I'm a student (I've been studying mathematics 4 years at the university) and I like functional analysis and topology, but I only studied 6 credits of functional analysis and 7 in topology (the basics)....
3 votes
1 answer
1k views

Borel-Cantelli lemma for general measure spaces (those with infinite measure)

The Borel-Cantelli lemma is often stated for a probability space or spaces with finite measure. But it seems to me that it still holds if the space $X$ is of infinite measure. I seem to be able to ...
6 votes
0 answers
189 views

Energy of harmonic maps from $\mathbb R^2$ to $S^2$ is quantized

Assume that $U:\mathbb R^2\to S^2=\{y\in\mathbb R^3:|y|=1\}$ is a smooth solution of the equation $\Delta U+|\nabla U|^2U=0$ in $\mathbb R^2$ with $\int_{\mathbb R^2}|\nabla U|^2\,dx<+\infty$. ...
0 votes
0 answers
72 views

Ultraproduct reflexive

Hello I have the following construction: Let $(E,\|\cdot\|_E)$ be a Banach space, $E_n:=E^n$ and $\|x_n\|_n:=\frac{1}{n}\sum_{k=1}^n \|x_n(k)\|_E$ for all $x_n=(x_n(1),...,x_n(n))\in E^n$ and $n\in\...
1 vote
1 answer
585 views

Integral on level sets

Let $g_\epsilon : K \subset \mathbb{R}^d \rightarrow \mathbb{R}$ (more regularity can be assumed if necessary) be defined on a compact set (with regular boundary) $K \subset \mathbb{R}^d$, and the ...
67 votes
3 answers
11k views

Nonconvexity and discretization

Edit: Here's a more down-to-earth, and somewhat weakened, but I believe still nontrivial, version of the main theorem. Prototypical nonconvex spaces are $\ell^p$-spaces for $0<p<1$, say $\ell^p(\...
2 votes
0 answers
120 views

Convergence of eigenfunctions

In their 1999 paper "Sturm-Liouville operators with singular potentials", Savchuk and Shkalikov prove the uniform resolvent convergence of an operator $L_\varepsilon \rightarrow L$ for $\...
4 votes
1 answer
702 views

Bound the operator norm of the Fréchet derivative of a Lipschitz function in this setting

I want to find a bound for the operator norm of the Fréchet derivative of a Lipschitz continuous function in the following setting: Let $E$ be a $\mathbb R$-Banach space; $v:E\to[1,\infty)$ be ...
4 votes
1 answer
164 views

Large JN-sets in Banach spaces

For every infinite-dimensional Banach space $X$ there is a weak*-null sequence in the unit sphere of $X^\ast$. Does this extend under suitable circumstances to the non-separable setting? Say that $X$ ...
1 vote
1 answer
116 views

Points in the Stone Cech compactification are intersection of open sets

Let $\beta \mathbb{N}$ be the Stone Cech compactification of the natural numbers and let $ x\in \beta \mathbb{N}$. Is it true that there exists a sequence of open sets $\{U_n\}_{n=1}^\infty$ in $\beta ...
3 votes
1 answer
162 views

Can homomorphisms be Borel in some weaker topology?

Let $X$ be a Banach space with separable dual and let $A$ be a Banach algebra. Consider a norm continuous homomorphism $h$ from $L(X)$, the Banach algebra of bounded operators on $X$ onto $A$. In $L(X)...

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