Questions tagged [fa.functional-analysis]

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

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5 votes
1 answer
178 views

Decomposition of non negative Radon measure into $L^1$ and $H^{-1}$ functions

What is a reference for the following result (which appears to be well-known in measure theory)? Any non negative Radon measure can be decomposed uniquely into the sum of an absolutely continuous ...
1 vote
2 answers
209 views

Halmos recurrence theorem for a locally compact group

The recurrence theorem of Halmos is well known in the case of a non-singular endomorphism $T$ of a measured space $(X,\mathcal B,\mu)$. A measurable subset $A$ is contained in the conservative part (...
19 votes
1 answer
602 views

The dual of $\mathrm{BV}$

$\DeclareMathOperator\BV{BV}\DeclareMathOperator\SBV{SBV}$I'm going to let $\BV := \BV(\mathbb{R}^d)$ denote the space of functions of bounded variation on $\mathbb{R}^d$. My question concerns the ...
10 votes
1 answer
482 views

Making sense of the formula $\operatorname{Det} (I+M )= e^{\operatorname{Tr} \ln (I+M)}$, especially in the infinite dimensional cases

$\DeclareMathOperator\Det{Det}\DeclareMathOperator\Tr{Tr}$In physics literature dealing with quantum field theory, the formula \begin{equation} \Det(I+M) = e^{\Tr \ln(I+M)} \end{equation} appears ...
1 vote
0 answers
42 views

Measure preserving maps of pseudo-Lebesgue measure in infinite-dimensional vector space

Let $I =([0,1),\mathcal{B},\lambda)$ stand for the unit interval with a Lebesgue measure constrained on it. This is just a uniform probability distribution, an infinite power $I^{\infty}$ is a well ...
0 votes
0 answers
45 views

Is identity map on the space of smooth maps smooth?

I'm curious about the identity map on the space of all smooth maps (between two locally convex topological vector spaces in the sense of convenient calculi) when we equip the space with different ...
0 votes
0 answers
31 views

Verifying the Proof of a Bilinear Estimate in $L^2$

$\newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\ds}{\displaystyle} \newcommand{\Lpn}[2]{\left\lVert#1\right\rVert_{L^{#2}}}$ $\newcommand{\Lptxy}[3]{\left\lVert#1\right\rVert_{L^{...
1 vote
1 answer
128 views

Finding the set of best approximation

Given $X$=$l^1$ and its dual space $X^*=l^\infty$. Now take $f=(1, 1/2, 2/3, 3/4,...) \in X^*$. Then clearly $\|f\|_\infty = 1$. I have found that $H=\ker f$ is a proximinal hyperplane in $X$. Note: A ...
0 votes
0 answers
76 views
+100

Approximate solution of the heat equation

I'm reading the article "Singularity formation for the two dimensional harmonic map flow into $S^2$" from J.Davila, M.del Pino and J.Wei and at some point, there are some computations I don'...
2 votes
0 answers
30 views

Generalizing Kato-Seiler-Simon-type inequalities to diamagnetic operators

I recently learned about estimates one can perform with operators on $L^2(\mathbb{R}^n)$ given as $f(x)g(-i\nabla)$, see Chapter 4 in Trace Ideals and their Applications by Professor Barry Simon (the ...
6 votes
1 answer
135 views

What happens if we consider functions of bounded variation that are not in $L^1$?

A function $f \in L^1(\mathbb R^n)$ is said to be of bounded variation if there exists a constant $C \geq 0$ such that $$ \int_{\mathbb R^n} f(x) \operatorname{div} \phi(x) \; dx \leq C \sup_{ x \in \...
3 votes
1 answer
1k views

Relation between two different definitions for relative sequential compactness

Building upon this question in Math.SE, I think the following might be rather of interest for MO. In the literature on measure theory, probability and functional analysis the definition of a subset $...
0 votes
0 answers
20 views

Characterization of the adjoint of a $C_0$-Semigoup infinitesimal generator

I am looking for characterizations of the adjoint operator of the infinitesimal generator of a $C_0$-semigoup in Hilbert spaces. All I could find in the literature is that if $(A, D(A))$ is the ...
2 votes
0 answers
165 views

Research in analysis of PDEs

I am currently figuring out what topic to work on for my undergraduate thesis and was able to narrow it down to mathematical analysis. As of now, I have two main options: a thesis working on Sobolev ...
1 vote
0 answers
62 views

Cyclic group action and finite invariant set

Let $(X, d)$ be a compact metric space and $G$ a discrete group acting on $X$ such that, for each $g\in G$, the mapping $x\mapsto g\cdot x$ defines a homeomorphism on $X$ Is it true that the ...
0 votes
1 answer
84 views

eigenvalues of matrices (with positive entries)

I am reading an old paper by Kawpien and Pelczynski, Studia Math. 1970. It claims that singular values of a matrix (with positive entries? I am not sure) is given by $t_i=\sqrt{\sum_{j\ge 1}a(i,j)^2}$....
0 votes
0 answers
37 views

What will be the Frobenius norm of Jacobian of inner product of two matrices?

I have a function called $h$. $$h=\langle\sigma(B_1s),\sigma(T_1p)\rangle$$ Here $B_1$ and $T_1$ are two matrices. $s$ and $p$ are vectors of $d_1$ and $d_2$ dimensions, respectively. $\sigma$ = ...
8 votes
1 answer
1k views

Chain rule for distributional derivative

Let $V \subset H \subset V^*$ be a Gelfand triple (eg. $H^1 \subset L^2 \subset H^{-1}$). Let $u \in L^2(0,T;V)$ have a distributional derivative $u' \in L^2(0,T;V^*)$. So $\int_0^T u(t)\varphi'(t) = ...
-1 votes
1 answer
179 views

A commuting pair of isometries

Let $H$ be a Hilbert space and $B(H)$ be the space of all bounded operators on $H$. The Wold decomposition says that: an operator $x$ in $B(H)$ is an isometry if and only if $x=x_u\oplus x_s$ where $...
2 votes
2 answers
218 views

Derivative of the absolute value

Let $f \in W^{1,p}(U)$, then how to prove that $|f| \in W^{1,p}(U)$, where $W$ means the sobolev space over some open subset $U \in \mathbb{R}^n$. In Lieb's Analysis he prove that Let $f$ be in $W^{1,...
1 vote
0 answers
41 views

Does an isometric automorphism of $L_p (X,\mu, E)$ preserve pointwise convergence?

Let $(X, \mathcal A, \mu)$ be a $\sigma$-finite measure space and $(E, |\cdot|)$ a Banach space. Here we use the Bochner integral. Let $p \in [1, \infty)$ and $q \in (1, \infty]$ such that $p^{-1}+q^{-...
0 votes
0 answers
57 views

Does this order of conclusion make sense? [migrated]

I have a function of two variables which can be written as $$\DeclareMathOperator{\D}{d\!} K(u,v)=K(f(u),g(v)). $$ I'm aware that $K$=constant is equivalent to $\frac{\D K}{\D u}=\frac{\D K}{\D v}=0,$...
1 vote
0 answers
50 views

Name for natural norm on functions to non-linear targets

Is there a standard term for the quasi-norm $$\|f\|_{[k]}=\sum_{i=1}^k(\sup\|f^{(i)}\|)^{1/i}$$ ? It is useful due to the fact that it is reasonably compatible with post-composition by smooth ...
1 vote
0 answers
260 views

When Pelczynski's property (V*) forces (V) in the dual space?

Recently, I have been asked if are there any sufficient conditions for a Banach space with Pelczynski property (V*) to have dual with property (V). Since it was a long standing open problem, I guess ...
7 votes
1 answer
244 views

Uniqueness of solutions of Young differential equations

Consider the following one dimensional Young differential equation: $$ Y_t=\int_0^t Y_s dX_s,\quad t\in[0,1]. $$ Here the driving process $X$ is a bounded functions $[0,1]\to\mathbb{R}$, which is $\...
4 votes
4 answers
4k views

General procedure for inverse of an integral transform

Is there a general inversion formula or procedure for an integral of the form (where f is the function being transformed and g depends on the type of transform) $\int^{a}_{b} f(x) g(x,\xi) dx $ ? ...
0 votes
0 answers
94 views

How to prove that $\bigotimes_2 X$ can be normed [migrated]

Let $X$ be a normed space. Then, $\bigotimes_2X=X\otimes X$ is also a normed space with the norm $$\|u\|=\inf\{\sum_{j=1}^n |\lambda_j| \|x_j\| \|y_j\| : u=\sum_{j=1}^n \lambda_jx_j\otimes y_j \},$$ ...
2 votes
1 answer
120 views

Function monotony between [0,T] and $L^2$

Let $\Omega\subseteq\mathbb{R}^N$ be a bounded and smooth domain. If $z:[0,T]\to L^2(\Omega)$ is a function in $H^1([0,T],L^2(\Omega))$ with the property that $z'(t)(x):=z'(t,x)>0$ a.e. on $\Omega$ ...
0 votes
0 answers
99 views

Check the compactness of the set in $l^1$ [closed]

Check the compactness of the set $K = \big\{x=(x_n)\in l^1: |x_n|^2 \leq n^{-1}\sin(|x_n|)\big\}$ in $l^1$. If we consider the same question in $l^2$, then $$ \|x\|\leq\sum\limits_{n=1}^\infty 1/n^2, $...
4 votes
2 answers
140 views

Reference request: "Tangent relation" in metric spaces

Let $X,Y$ be metric spaces. Let $f,g : X \to Y$ be two maps and $x_0 \in X$. Let us say that $f$ and $g$ are tangent at $x_0$ if the following condition is satisfied: For every $\epsilon > 0$ there ...
2 votes
0 answers
35 views

A Lipschitzian's condition for the measure of nonconvexity

I'm actually working on the measure of nonconvexity and its application. Especially, the Eisenfeld–Lakshmikantham MNC defined - in a Banach space - by: $$\alpha(A)=\sup_{b\in\overline{\operatorname{...
1 vote
1 answer
1k views

Monge–Ampère operator

I'm studying the article of Bedford–Taylor "Fine topology, Šilov boundary…" but I don't understand the proof of the following proposition. Let $u$, $v$ be plurisubharmonic functions defined ...
1 vote
0 answers
34 views

A question of interpolation space on homogeneous Carnot group

Let $\mathbb{G}$ be the homogeneous Carnot group on $\mathbb{R}^{n}$ defined as follows: A homogeneous Lie group $\mathbb{G}=(\mathbb{R}^{n},\circ)$ is called a homogeneous Carnot group (or a ...
6 votes
0 answers
92 views

Heat Flows and spatial singularities

While working on an abstract problem, I came up with the following question: Let $\Omega_1 := \mathrm B(-1, 1)$ and $\Omega_2 := \mathrm B(1, 1)$, where $\mathrm B(x, r) \subseteq \mathbb R^2$ denotes ...
8 votes
4 answers
485 views

Uniform density of Lipschitz maps is space of continuous function — for general metric spaces

Let $X$ and $Y$ be metric space, $X$ be compact, $C(X,Y)$ denote the set of continuous functions from $X$ to $Y$ with uniform convergence on compacts topology, and $\operatorname{Lip}(X,Y)$ denote the ...
4 votes
1 answer
97 views

A ball with slit at the radius is not $W^{1,1}$-extension domain

Recall that: A domain $\Omega\subset \mathbb{R}^d$ is an $W^{1,1}$-extension domain if there exists an operator $E:W^{1,1}(\Omega)\to W^{1,1}(\mathbb{R^d})$ and a constant $c= c(d,\Omega)>0$ such ...
1 vote
0 answers
70 views

Convergence by parameter involving inf

Suppose $y\in U$ is a parameter in some space $U\subset \mathbb{R}$ and we consider the following problem: $$ a(y)=\inf_{u\in X}\left\{ \int_{\Omega}|\nabla u|^2+\int_{\Omega} P(y)|u|^2 \right\} $$ ...
6 votes
1 answer
445 views

Fourier series of smooth functions in infinitely many variables

Let $J$ be a set (usually countable). Let $t_j$, $j\in J$, be variables in ${\mathbb R}/2\pi i{\mathbb Z}.$ Put $u_j=\exp(it_j),$ $j\in J.$ Introduce the following semi-norms on the space of Fourier ...
5 votes
1 answer
835 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 $B(X,Y)$?
1 vote
1 answer
176 views

Hölder continuity of Radon transform of smooth function

Given an integrable function (e.g a probability density function) $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined by $$ R[f](w,b) := \int_{\mathbb R^n} \delta(x^\top w - b)f(x)...
5 votes
1 answer
120 views

On existence of a concave function

Let $a$ be a strictly positive $C^\infty$ smooth function on the unit interval. Does there exist a strictly positive $C^\infty$ smooth function $f$ on $I$ such that $$ f’’(x) \leq 0\quad \text{and} \...
35 votes
2 answers
11k views

What, exactly, has Louis de Branges proved about the Riemann Hypothesis?

Hi, I know this is a dangerous topic which could attract many cranks and nutters, but: According to Wikipedia [and probably his own website, but I have a hard time seeing exactly what he's claiming] ...
4 votes
0 answers
129 views

About the structure of smooth automorphic forms

Recently I read Prof. Cogdell's notes: Lectures on L-functions, Converse Theorems, and Functoriality for $GL_n$. (Co) In chap.2.3, the conception of smooth automorphic forms is introdued. Explicitly, ...
2 votes
1 answer
172 views

Spectrum of $(Jx)_n =i((2n+1)x_{n+1}-(2n-1)x_{n-1})$ on $\ell^2(\mathbb{Z})$

I've been working on the spectrum of the closure of the operator $J: \mathcal{D}(J)= \mbox{span}\{ e_n: n \in \mathbb{Z}\} \subseteq \ell^2(\mathbb{Z}) \to \ell^2(\mathbb{Z})$ defined for $x=(x_n)_{n \...
4 votes
2 answers
158 views

Is the conditional expectation of a Caratheodory function a Caratheodory function?

Let $(Y, \Sigma,\mu)$ be measure space and $X$ a Polish space endowed with its Borel $\sigma$-algebra. Suppose that $f:Y\times X\to \mathbb R$ is a Carathéodory function (i.e. continuous in $x\in X$ ...
1 vote
0 answers
60 views

Fundamental Solution to Biharmonic Equation in 3D

(This is a repost of a question posed in StackExchange that didn't get any replies.) Is anything known about the fundamental solution to the equation: $$\nabla^4 (Au) + \nabla^2 (Bu)+Cu=0$$ for ...
1 vote
0 answers
83 views

How to prove a concentration isoperimetric inequality for a non-Lipschitz function

Definition $1$. A probability measure $\mu$ on $\mathbb{R}^{d}$ satisfies c-isoperimetry if for any bounded L-Lipschitz $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$, and any $t \geq 0$, \begin{align} \...
0 votes
0 answers
60 views

How are eigenvalues of two psd kernels related?

Suppose $K(x,x')$ and $R(y,y')$ are two positive semi-definite kernels on $(x,x')\in \mathbb {X}\times\mathbb{X}$ and $(y,y')\in\mathbb{Y}\times\mathbb{Y}$, respectively, and satisfying the following ...
4 votes
1 answer
156 views

Understanding the Osterwalder-Schrader conditions as formulated by Glimm and Jaffe

$\newcommand{\real}{\mathrm{real}}$I am having trouble with understanding the axiom (OS3) in this book by Glimm and Jaffe. It defines \begin{equation} \mathcal{A} = \left \{ A(\phi) = \sum_{j = 1}^N ...
0 votes
0 answers
63 views

Operators decomposition in pseudo-Hilbert space

Let $(H,g)$ be a pseudo-Hilbert space, i.e. $H$ is an infinite dimensional vector space endowed with an indefinite symmetric product $g$. Suppose we have a linear operator $D:H\to H$ and let $D^*$ be ...

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