# Questions tagged [fa.functional-analysis]

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

7,675
questions

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### If $f \circ u \in BV$ and $f$ is strictly monotone, then is $u \in BV$?

Let $f: \mathbb R \to \mathbb R$ be a Lipschitz strictly monotone (so, in particular, invertible) function. Let $u: \mathbb R \to \mathbb R$. If $f \circ u \in BV$ can we conclude that $u \in BV$?

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71 views

### Orthogonal decomposition of $L^2(SM)$

I have been stuck on the following problem for a long time but I could not get the answer. Would you please help me? I was reading one paper [Paternian Salo Uhlmann: Tensor tomography on the surface] ...

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221 views

### Can Birkhoff's ergodic theorem for integrable functions easily be deduced from Birkhoff's ergodic theorem for bounded functions?

It seems to me that a considerably simpler proof [see below] of Birkhoff's ergodic theorem can be obtained for bounded observables than for more general $L^1$ observables. Therefore, I feel like it ...

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249 views

### A completely positive equivariant map $\varphi: A \to B$ induces a map $A \rtimes_r \Gamma \to B \rtimes_r \Gamma.$

Recall the construction of the reduced crossed product:
Let $\Gamma$ be a discrete group and $A$ be a $C^*$-algebra with an action $\alpha: \Gamma\to \operatorname{Aut}(A)$. Consider the $*$-algebra $...

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468 views

### If I multiply the coefficients of a trace-class operator with bounded complex numbers is it still trace class?

Suppose that $T \in TC(l^2( \mathbb{Z}))$ is trace class.
Consider its kernel $ T(i,j) = \langle e_i, T e_j \rangle $ where $ \{e_i\}_{i \in \mathbb{Z}}$ is an ONB for $l^2( \mathbb{Z})$. Now, ...

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52 views

### eigenvalues of integral operator with centered kernel

Suppose $k:\mathcal{X} \times \mathcal{X} \to \mathbb{R}$ be a symmetric positive (semi-)definite kernel. The Moore-Aronszajn Theorem indicates that
there is a reproducing kernel Hilbert Space $\...

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90 views

### Confusion on the paper "Cohomology of maximal ideal space"

In the paper Cohomology of Maximal Ideal Space, there is a corollary about if $M$ is a compact orientable n-dimensional manifold, then $C(M,\mathbb{C})$ cannot be generated by fewer than n+1 elements.
...

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161 views

### Reference for Choquet-like theorem

While reading a paper, I encountered the following statement:
Let $K$ be a convex compact subset of a locally convex topological vector space. If $\mu \in P(K)$ is a Radon probability measure on $K$, ...

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111 views

### Techniques for showing non-degeneracy results (PDE)

Motivation:
Consider the equation,
$$-\Delta u = u^p$$
in $\mathbb{R}^n$ for $n\geq 3$ and $p=2^*-1.$ Then we know that this equation has unique positive solutions given by functions of the form $U_{a,...

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93 views

### Norm estimate for the difference between a positive operator and its expectation

Let $C_p$, $1<p<\infty$, be the Schatten-$p$-class.
Let $x\in C_p$ be positive and $\|x\|_p =1$.
Let $E$ be the conditional expectation onto the diagonal part.
If $\|E(x)\|_p \ge 1-\delta $ for ...

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88 views

### Cech cohomology on product covers & Fréchet sheaves

My question is about the paper [Ka67].
Let $S, T$ be sheaves of nuclear Fréchet spaces over paracompact topological spaces $X, Y$, respectively; in particular, if $V \subset U$ are open subsets in $X$,...

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30 views

### $L^p$-continuity for discrete linear causal systems

Let $p \in [1, +\infty)$, $(b_0(n)), \dots (b_m(n)), (a_1(n)), \dots, (a_m(n))$ suitable sequences of real numbers and consider the map $\phi: \ell^p \to \ell^p$, $x \mapsto y$ defined by:
\begin{...

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66 views

### On a core for Neumann Laplacians

Let $D \subset \mathbb{R}^d$ be a bounded smooth domain. We consider the Neumann semigroup $\{T_t\}_{t>0}$ on $C(\overline{D})$. In other words, $\{T_t\}_{t>0}$ is the semigroup of the normally ...

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792 views

### Continuous linear functionals and the Axiom of Choice

Can one prove without the Axiom of Choice that for every normed vector space $X$ there exist a nonzero continuous linear functional on $X$?

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88 views

### Closure of $f\mapsto\sigma f''$ on $\mathcal{C}^2(\,[0,1]\,)$

Let $\sigma\in\mathcal{C}^0(]0,1])$ a positive function such that $\lim\limits_{t\rightarrow 0}\sigma(t)=0$, and $f\in\mathcal{C}^0(\,[0,1]\,)\cap\mathcal{C}^2(\,]0,1]\,)$ such that $\lim\limits_{t\...

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137 views

### 'Diamagnetic' inequality for negative Sobolev spaces

Let us look at the subspace of smooth complex functions of $L^2(\mathbb{R}^n,\mathbb{C})$, call $H^s$ the Sobolev spaces. By the diamagnetic inequality $\lvert \nabla \lvert\psi\rvert\rvert (x) \le \...

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83 views

### Continuity of the entropy of the solution of a parabolic PDE at $t=0$

Consider the following initial value problem for a parabolic PDE :
$$\begin{cases}
\textrm{div}\big(A\,\nabla u(t,x) + b(x)\, u(t,x)\big) \,=\,\partial_t u(t,x) \quad x\in\mathbb R^d\,,\ t>0 \\[4pt]...

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21 views

### Spectrum of a linear operator depending of the fractional laplacian operator

Let $s \in \mathbb{R}$ such that $0<s<1$. Consider the periodic fractional Laplacian $(-\Delta)^s$ in the real line defined via Fourier series as follows: if $f:[-\pi,\pi] \subset \mathbb{R} \...

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107 views

### Definition of a $\psi$-Banach space [closed]

Let $X$ be a Banach space. Let $\mathcal{F}$ be the family of all the bounded subsets of $X$. If $I$ is the identity map on $X$, we shall denote by $\operatorname{span}\{I\}$ the vector space ...

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131 views

### Dual Banach space $X^*$ complemented in $\mathrm{Lip}_0(X)$?

$\DeclareMathOperator\Lip{Lip}$Let $X$ be a real Banach space. The dual $X^*$ is a closed subspace of $\Lip_0(X)$. ($\Lip_0(X)$ denotes the space of real-valued Lipschitz functions $f:X\to\mathbb{R}$ ...

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51 views

### Two estimates in the Sobolev spaces

Let $\mathbb{T}^d$ be the $d-$dimensional torus and $u: \mathbb{T}^d \to \mathbb{C}$ be a complex valued function. Denote $$f(u) = |u|^p u,$$ where
$1 \leq p \leq \frac{4}{d-2}$ if $d\geq 3$ and $1 \...

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109 views

### Conditions replacing compactness

Reading this book, the authors used the following "classic" idea:
Let $X$ be a Banach space and $C$ a nonempty, weakly compact, convex subset of $X$. Let $T: C \rightarrow C$ be a ...

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108 views

### Geometry in Hilbert spaces / spheres in high dimensions

Let $H$ be a separable Hilbert space of infinite dimension and let $(e_n)_{n \in \mathbb{N}}$ be an orthonormal basis of $H$. For a series $(\alpha_n)_{n \in \mathbb{N}} \subset \mathbb{R^+}$ we are ...

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323 views

### Condensed/liquid vector spaces and path integrals

[Edited to take into account comments.]
Background
One approach to the problem of making rigorous various measures on spaces of paths (for example, the Wiener or Feynman measure) is the time-slicing ...

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76 views

### Densely defined derivations in von Neumann algebra(in norm topology)

This post is actually a refined question of here.
Let $N$ be an abelian von-Neumann algebra. Define densely defined derivation to be derivation $\delta:D(\delta)\rightarrow N$ where the domain is ...

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63 views

### Does $L^{\infty}[0,1]$ admits infinitely many densely defined derivations in weak* topology?

To clarify the question. First we define what is densely defined derivation. A densely defined derivation $\delta:D(\delta):\rightarrow L^{\infty}[0,1]$ where $D(\delta)$ is a dense subalgebra( in ...

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423 views

### A little problem in PDE or function analysis

Let
$E$ be the usual sobolev space $H^{1}_{0}(\Omega)$ on a smoothly bounded domain $\Omega$,
$E_{k}$ be its subspace spanned by the first $k$ eigenfunctions of the Laplace operator, i.e.
$$E_{k}:=\...

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40 views

### Quartic vs. quadratic exponential on locally convex spaces

Suppose we have a (Radon) locally convex space $X$ with a centred Gaussian measure $\mu$ and two continuous seminorms $p$ and $q$ on $X$.
Under what circumstances could we expect
\begin{equation}
\...

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141 views

### Proof: If a reproducing kernel exists for a Hilbert space, then it is unique

I really want to prove the statement in the title but I'm struggling with it. Here my current state:
Proof via contradiction. Let $\mathcal{H}$ be a RKHS with two reproducing kernels $k$ and $\hat{k}$ ...

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82 views

### Results on the eigenspace of weighted elliptic eigenvalue problems

I am considering the following eigenvalue problem in $\Omega=\mathbb{R}^n_+$
$$-\operatorname{div}(a(x)\nabla \varphi) = \lambda a(x)w(x)\varphi$$
where the weights $a>0$ and $w\in L^{\infty}$ (and ...

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130 views

### Questions about Maharam's classification theorem

I am studying von Neumann algebras. In the wiki article abelian von Neumann algebras, it mentions that every abelian von Neumann algebras acting on a separable Hilbert space is *-isomorphic to $L^{\...

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88 views

### Converse of mean value theorem almost everywhere? (Version 2)

Note: This is an attempt to narrow down conditions under which the conjecture stated in this previous post is true. As stated, it is false as shown by the counterexample provided in the answers by the ...

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39 views

### Nested nets of closed bounded star-shaped sets in a semi-reflexive space

Among Hausdorff locally convex spaces, semi-reflexivity is characterized by the weak topology having the Heine-Borel property. It follows that, in a semi-reflexive space, every nested net of closed ...

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304 views

### Is Hausdorffness a categorical property in the category of locally convex spaces?

I want to characterize Hausdorffness of a locally convex space only using categorical terms of the additive category LCS of locally convex spaces and continuous linear maps, i.e., terms like mono- or ...

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99 views

### Semi-linear elliptic problem, energy functionals, Fréchet derivatives and the Newton method in Banach spaces

Suppose $\Omega\subset\mathbb{R}^n$ is a regular open set, $f\in L^2(\Omega)$ and consider the following elliptic problem.
$$-\Delta u + u=f'(u) , \;\;u_{|\partial \Omega}=0,$$
where $f'$ is the ...

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53 views

### Is the lattice of bounded Henstock Kurzweil integrable functions countably complete?

The set of HK integrable functions with an integrable upper bound $f$ forms a lattice, and satisfies the MCT and DCT. Does this mean that the lattice is countably complete?
Indexing any countable set, ...

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100 views

### Density of traces of solutions to an elliptic equation

Let $D_1$ be a domain with smooth boundary and assume that $D_1$ is a proper subset of $D_2$ which is itself a bounded domain in $\mathbb R^n$ with a smooth boundary. Assume also that $D_2\setminus ...

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88 views

### Books on limiting properties of matrices with growing size

This question has been posted on Math-Se previously.
I am studying asymptotic properties of the Projection Matrix
$$
H_n=X'(X'X)^{-1}X
$$
By the Gerschgorin disc theorem, the bounds on the ...

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84 views

### Equivalence between notions of dynamical coupling as defined by Villani in his book Optimal Transportation: Old and New

$\DeclareMathOperator\law{law}$In Villani's book he presents the following notions of dynamical couplings:
Let $(X,d)$ be a Polish space. A dynamical transference plan $\Pi$ is a probability measure ...

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50 views

### Characterization of inverse limits of finite-dimensional convex cones

Consider a countable inverse system $C_1\substack{f_1 \\ \leftarrow} C_2 \substack{f_2 \\ \leftarrow} C_3 \substack{f_3 \\ \leftarrow} \ldots$ where the $C_i$ are finite-dimensional convex cones of ...

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156 views

### Set where the speed of convergence is uniform in Lebesgue's density theorem

Let $B \subset \mathbb R^n$ be the unit ball.
Consider a Borel measurable set $E \subset B$ with positive Lebesgue measure $|E|>0$ (say $|E| = |B|/2$).
Then, Lebesgue's density theorem, says that ...

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679 views

### Criterion for compactness

Let $T:H\to H$ be a continuous operator on a Hilbert space.
Assume there exists an orthonormal base $(e_j)_{j\in\mathbb N}$, such that the sequence $Te_j$ tends to zero.
Must $T$ be compact?

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61 views

### When does an RKHS contain another?

Consider a psd kernel function on the unit-sphere in $\mathbb R^d$ off the form $K(x,x') = \varphi(x^\top x')$ for some $\varphi:[-1,1] \to \mathbb R$, and let $\mathcal H_\varphi$ be the induced ...

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108 views

### Proving an estimate for the Neumann problem on $\mathbb{R}^3 \setminus B_1$ in Weighted Sobolev spaces

Let $M := \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball.
It is known that for every $g \in H^{\frac{1}{2}} (\partial M)$, and for an appropriate $\delta$, there exists a unique solution $u$ ...

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68 views

### Reference request: Optimal controls can be assumed to take values in a convex set

Consider the deterministic controlled system:
$$\dot x(t) = Ax(t) + Bu(t), \ t \in [0, T]$$
$$x(0) = x_0$$
where $x: [0, T] \to \mathbb R^n$ is the controlled state process, $A \in \mathbb R^{n \times ...

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41 views

### Convergence in the resolvent sense and spectral properties

Let $\{T_k\}_k, T$ be unbounded selfadjoint operators on a Hilbert space $H$. If $T_k\to T$ in the norm-resolvent sense, then for any $(a,b)\subset \mathbb R$ with $\{a,b\}\cap \sigma(T)=\emptyset$, ...

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167 views

### Schauder basis in the Arens-Eells space

Context
Arens-Eells space. Let $X$ be a separable pointed metric space with base point $e$. An elementary molecule is defined as follows (Nik Weaver, Lipschitz Algebras, 2nd ed.)
$$
m_{pq} := \delta_p ...

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48 views

### Method of characteristics and explicit formula for an IBVP for the transport equation

Let us assume $c:(0,1) \to (\epsilon,+\infty)$ and consider the PDE
$$
\begin{cases}
u_t+c(x)u_x = 0, \\
u(0,x) = g(x) \\
u(t,0) = f(t)
\end{cases} \qquad (t,x) \in (0,T)\times(0,1) \label{1}\tag{$\...

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112 views

### Uniqueness of the predual of a W*-algebra

Consider the following fragments in Takesaki's "Theory of operator algebras" (volume I):
Question: So, we have an abstract Banach space $F$ with $A \cong F^*$. In Lemma 3.6, one considers ...

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415 views

### Unbounded operators vs compact operators

The operator $L:\operatorname{dom}(L)\subset C[0,1]\to C[0,1]$ given by $Lx=x'$
a) is closed, unbounded and densely defined
b) also has a compact right inverse, namely $K:C[0,1] \to C[0,1]$ given by $...