# Questions tagged [fa.functional-analysis]

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

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

8,510
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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 ...

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2
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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 (...

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$\DeclareMathOperator\BV{BV}\DeclareMathOperator\SBV{SBV}$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 ...

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

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0
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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 ...

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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 ...

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

1
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1
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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 ...

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+100

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'...

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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
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1
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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
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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 $...

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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 ...

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0
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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 ...

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0
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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
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1
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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 $t_i=\sqrt{\sum_{j\ge 1}a(i,j)^2}$....

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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. $\sigma$ = ...

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1
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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) = ...

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1
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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

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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,...

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0
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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^{-...

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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
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0
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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
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0
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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
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1
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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

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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
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0
answers

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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
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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
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0
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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
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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
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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

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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
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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
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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 ...

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votes

4
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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
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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
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0
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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
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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

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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
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1
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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

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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
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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
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0
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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

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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
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2
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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

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(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

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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
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0
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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
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$\newcommand{\real}{\mathrm{real}}$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
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0
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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 ...