# Questions tagged [fa.functional-analysis]

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

5,346 questions

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41 views

### Transformation of Fourier Transform

Suppose that $f$ is a function with a Fourier transform, and that $g:\mathbb{R}\rightarrow \mathbb{R}$ is a smooth function such that $g\circ f$ has a Fourier transform also.
Is there an expression ...

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**1**answer

136 views

### Can smoothness of curves into a convenient locally convex vector space be tested with just a dense subspace of the dual?

Let $E$ be a (Hausdorff) locally convex vector space (from now on just "lcs" for short). We say that $E$ is convenient (also called locally complete, Mackey-complete or $c^\infty$-complete) if, given ...

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82 views

### Growth rate of bounded Lipschitz functions on compact finite-dimensional space

Let $\mathcal X$ be a metric space of diameter $D$ and "dimension" (e.g doubling dimension) $d$. Let $L \in [0, \infty]$ and $M \in [0, \infty)$ and consider the class $\mathcal H_{M,L}$ of $L$-...

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956 views

### Equivalence of fractional Sobolev space defined through Gagliardo norm and interpolation; dependence on the domain

Let $\Gamma$ be a smooth hypersurface in $\mathbb{R}^n$. We can define the fractional Sobolev space
$$X = \{ u \in L^2(\Gamma) \mid |u|_X^2 := \int_\Gamma \int_\Gamma \frac{|u(x)-u(y)|^2}{|x-y|^{n+1}} ...

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173 views

### Sufficient criteria for $X \subset \mathcal{H}$ to be a Lipschitz (or unif. cont.) retract of $\mathcal{H}$

I am interested in sufficient criteria which ensure that a subset $X$ of a Hilbert space $\mathcal{H}$ is a Lipschitz (or at least uniformly continuous) retract of $\mathcal{H}$.
Under which ...

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26 views

### Potential for a Monotone Operator

[Cross-posted from math.stackexchange]
I have a question about understanding the proof of Theorem 4.11 in the paper A Potential Theory for Monotone Multivalued Operators (accessible here). The ...

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**1**answer

126 views

### Smooth structure on the space of sections of a fiber bundle and gauge group

Let $\xi$ be a fiber bundle $F\hookrightarrow E\to B$ (where every space is smooth, T2 and second countable), let $\Gamma(\xi)$ be the space of smooth sections. We can complete $\Gamma(\xi)$ with ...

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28 views

### Optimal transport between Gaussian mixtures and their centers

I have a question about bounding the Wasserstein loss between a continuous Gaussian mixture and a discrete uniform distribution of its centers. In particular, let $P=\frac 1 k \sum_{i=1}^k \mathcal{N}(...

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58 views

### Conditions for the embeddig of the space $L^\infty(I, W^{1,2}(U))$ into $L^\infty(I \times U)$

Let $I$ be a compact interval of $\mathbb{R}$ and $U$ be a bounded subset of $\mathbb{R}^2$.
If $f \in L^\infty(I, W^{1,2}(U))$, what (non-trivial) condition ($L^p$-estimate on $f$ or decay-like ...

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210 views

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### Meaning of Alberti rank-one theorem

Heuristically what does Alberti's rank-one theorem imply about the structure of a $\mathrm{BV}$ vector field $\boldsymbol{b}$?
Is it rigorously fair to say that the level lines of $\boldsymbol{b}$ ...

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70 views

### Hilbert-Schmidt Operators

I am reading Brezis's book : Functional Analysis, Sobolev Spaces and Partial Differential Equations and trying to solve an exercise on Page 499. The question is as follows: Assume that $\Omega = (0, 1)...

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124 views

### Computing Bohr Radii

The Bohr radius $R$ for $\mathcal{H}(\mathbb{D})$ is defined as $$R = \sup\limits_{0<r<1} \left\{ r\ |\ \sum\limits_{k=0}^{\infty}|a_k|r^k \leq |f|_\mathbb{D} \text{ for all }f(z)=\sum\limits_{k=...

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81 views

### Semi-continuous fields of C*-algebras having dimension one on a dense set

Given a Hausdorff, locally compact space $X$, let us consider a semi-continuous field $\{A_x\}_{x\in X}$ of C*-algebras over $X$, such that $A_x$ is one-dimensional for every $x$ in a dense subset $D$ ...

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80 views

### Examples of non-isomorphic $C^\ast$ algebras with isomorphic quasi-state spaces

Let $A$ (resp. $B$) be a unital $C^\ast$-algebra, $\mathcal{Q}(A)$ (resp. $\mathcal{Q}(B)$) the compact convex subset of $A^\ast$ equipped with the $\sigma(A^\ast, A)$ (resp. $\sigma(B^\ast, B)$) ...

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123 views

### On the continuity and injective-ness of Gauss quadrature scheme for numerical integration, with weight function identically $1$

Fix an integer $n\ge 2$. Let $[a,b]$ be an interval and $f: [a,b]\to \mathbb R$ be a continuous function and for $x_1,...,x_n$ being the Gaussian Quadrature nodes in $[a,b]$, and Gaussian Quadrature ...

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74 views

### Weak closure of subsets of the unitary sphere of a Banach space

Assume that $(X,\|\cdot\|)$ is a Banach space with $\|\cdot\|$ strictly convex. Define $S=\{x\in X:\|x\|=1\}$. Suppose that $\varepsilon>0$ and $x_0\in S$ and define
$$
B_\varepsilon=\{x\in X:\|x-...

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82 views

### Compact Embedding Between Parabolic Holder Spaces

My question is about the following compact embedding:
\begin{equation}
C^{\sigma+2, \sigma/2+1}_{x, t}(Q_T)\hookrightarrow C^{\sigma, \sigma/2}_{x, t}(Q_T).
\end{equation}
what condition should be put ...

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476 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 ...

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**1**answer

123 views

### For every table of interpolating nodes, there is a positive continuous function whose interpolating polynomials are not positive infinitely often

Fix an interval $[a,b]$. Is it true that for every table of interpolating nodes $\{x_{0,n},x_{1,n}...,x_{n,n}\}_{n=1}^{\infty}$, there exists a continuous function $f:[a,b]\to (0,\infty)$ such that ...

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366 views

### Efficiently reversing the triangle inequality with additional information

Suppose $f$ and $g$ are bounded functions, having whatever niceness properties you want, on some space of finite measure. Assume they are normalized so that $\int |f|^2=\int|g|^2=1$. I am looking for ...

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142 views

### Improving equi-integrability for a family $\mathcal F$ in $L^1(\Omega)$

Let $\mathcal F$ be a weakly compact subset of $L^1(\Omega)$. Dunford–Pettis theorem says that $\mathcal F$ is uniformly integrable. Also, by de la Vallée-Poussin theorem we can find an increasing ...

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95 views

### Interesting examples of spectral decompositions of BOUNDED operators with both continuous and discrete spectrum

I would like to have a few basic examples of bounded self-adjoint operators $T$ (more generally bounded normal would be fine) on a Hilbert space $(H,\langle,\rangle)$ for which the following criteria ...

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55 views

### Notions of $\beta$-Hölder smoothness when $\beta\in (1,2]$: are they equivalent?

I posted the following question on StackExchange a few months ago (https://math.stackexchange.com/questions/2898620/notions-of-beta-h%C3%B6lder-smoothness-when-beta-in-1-2-are-they-equivalent), but ...

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36 views

### Uniqueness of solution of Volterra Integral Equation with deviating argument

In the context of a physics problem, I am looking at a linear integral equation 2nd kind Volterra equation with deviating (centrosymmetric) argument in the unknown $u(t) \in L^2[a,b]$:
\begin{equation}...

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115 views

### Continuous embedding of the Skorohod space D(0,1) into L^2(0,1)

Let $D(0,1)$ be the Skorohod space with the Skorohod topology, i.e. the space of real-valued càdlàg-functions on $[0,1]$ with topology induced by the metric
$$d(f,g) = \inf_{\varphi \in \Lambda} \left\...

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116 views

### Taylor spectrum of commuting operators

Taylor spectrum of commuting operators
Fom the following paper (M. Ch—o, H. Motoyoshi, B. Na¡cevska Nastovska: On the joint spectra of commuting tuples of operators and a conjugation) we have
Let ...

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51 views

### Reference on vector-valued convex conjugate

The following definition of convex conjugate is taken from Wiki:
Let $X$ be a real topological vector space, and let $X^*$ be the dual space to $X.$
Denote the dual pairing by
$$\langle \cdot ,...

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62 views

### k-rank numerical range of an operator

Let $T\in\mathscr{B(\mathcal{H})}$ where $\mathcal{H}$ is an infinite dimensional seperable Hilbert Space and $k\in\mathbb{N}\cup\{\infty\}$. Now we define k-rank numerical range of $T$ denoted by $\...

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332 views

### What are the 'wonderful consequences' following from the existence of a minimal dense subspace?

In Peddechio & Tholens Categorical Foundations they quote PT Johnstone in their chapter on Frames & Locales:
...the single most important fact which distinguishes locales from spaces: the ...

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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 ...

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653 views

### solution uniqueness of non-linear Fredholm equations

the equation is
$F(x)=G\left(\int k(x,y)f(y)dy\right)$ $(*)$
where $f(x)=\frac{dF(x)}{dx}$ is unknown and it's required to be non-negative. With integration by parts we'll have the form of a non-...

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80 views

### On functions obtained from Gaussian Quadrature integration

Fix an integer $n \ge 2$. Let $x_1,...,x_n$ s and $w_1,...,w_n$ s be the Gauss Quadrature nodes and weights respectively in the interval $[0,1]$ (https://en.wikipedia.org/wiki/Gaussian_quadrature) . ...

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49 views

### Convolution with an analytic semigroup

Let $e^{At}$ denote an analytic semigroup on Hilbert space $X$ generate by $A:D(A)\to X$. Also, let $f\in L^1(0,\tau;X)$. I want to show that the convolution
$$ g(t)=\int_0^t e^{A(t-s)}f(s)ds$$
...

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129 views

### Parabolic variational inequality: regularity of the time derivative in $L^2(0,T;H)$?

Let $V \subset H \subset V^*$ be a Gelfand triple of Hilbert spaces. Take $f,\psi \in L^2(0,T;H)$ and consider the VI: find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V^*)$ such that
$$u(t) \leq \psi(t) ...

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339 views

### If the closed unit ball of Banach space has at least one extreme point, must the Banach space the be a dual space?

Let $X$ be a Banach space.
By Banach-Alaoglu and Krein-Milman Theorems, one can show that if $X$ is a dual space, then $X$ must have at least one extreme point of the closed unit ball.
I am ...

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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 ...

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2k views

### Elementary applications of Krein-Milman

This is a cross-post from MSE: Elementary applications of Krein-Milman. I'm starting to suspect that the question just doesn't really have a great answer, it's worth a try.
Recall that the Krein-...

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34 views

### Extension of a result about measurable, additive functionals

Let $W$ be a set, and let $v$ be a finitely additive probability measure on $2^W$.
Equip $2^W$ with the Borel sigma-algebra $\mathcal{B}$ generated by the sub-basic sets of the form $\{a: w \in a\}$ ...

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282 views

### Strong convergence of differential quotient in $L^2(0,T;V^*)$

I have got a problem regarding the weak differentiability of Bochner-integrable functions. Let $(V,H,V^*)$ be a Gelfand-triple and
\begin{align*}
w \in W(0,T) := \{w\in L^2(0,T;V) ~\vert~ \exists w' \...

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153 views

### Integration of hypergeometric product for legendre polynomials

I'm looking for a general solution to the integral:
$\int_{0}^1 {_2}F_1(m-n,m+n+1,m+1;z){_2}F_1(m-k+1,m+k+2,m+2;z) dz$
where $m,n,k\in \mathbb{N}$ and $m\leqslant n$ and $m+1 \leqslant k$.
To give ...

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45 views

### Existence of a function satisfying some integral conditions

I need help to prove the existence of a real function $h(x) \in C^1$ with condition that near zero $h(x) \sim \ln(x)$ and near infinity $\lim h(x)_{x \to \infty} = \infty$ such that following ...

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108 views

### Relation between Optimal Transport Cost and Difference between Topological Invariants?

I was working on some mathematics of Wasserstein GAN and found out a seemingly interesting research problem but I am not quite sure whether it has already been studied in some recent literature of ...

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127 views

### “Compactness in Measure” in Function Spaces

In Chapter 4.9 of the book "Measures of Noncompactness and Condensing Operators" (Vol. 55 of Operator Theory: Advances and Applications), the authors mention the property "compactness in measure". ...

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93 views

### Is this basis a Schauder basis?

Let $G$ be a torsion free group. Let $\alpha$ be an element in $\mathbb CG$, the group algebra of $G$, with $\|\alpha\|_1=1$ and assume that
$\{1,\alpha,\alpha^2,\dotsc\}$ is linearly independent,
$(...

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135 views

### Riesz representation theorem for Hilbert-to-Hilbert mappings [closed]

Assume $\phi:\mathbb{H}_1\rightarrow \mathbb{H}_2$ is a continuous linear mapping between two real Hilbert spaces $\mathbb{H}_1$ and $\mathbb{H}_2$. If $\mathbb{H}_2=\mathbb{R}$, then the Riesz ...

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245 views

### The tightest upper bound on $-\left\langle B(y-x),\nabla f(A x)\right\rangle$

Let $f$ be a $C^2$ convex function with Hessian $H$ such that $lI\mathcal\le H(x)\le LI,\ \forall x\in\mathbb{R}^n$. Then using Taylor's theorem it follows that $$f(x)-f(y)+\frac{l}{2}\|y-x\|_2^2\le-\...

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176 views

### A measure of noncompactness by a convex function

Let $E, \left \| \right \|$ be a Banach space, $\mathfrak{M}_E$ indicate a family of all nonempty bounded subset of $E$, $\mathfrak{N}_E$ the familly of all relatively compact sets, and $Ker \mu=\{X\...

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41 views

### Can we extract information from signature (rough path theory) to construct part of signal?

This question is related to rough path theory. Consider we have obtained signature obtained from a set discrete data points postulating linear from one data point to another. Such signature are used ...

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59 views

### Continuity of the positive part of a linear functional

Let $X$ be a vector space containing a convex cone $C$ which induces a (pre) order on $X$ by $x \geq y \Leftrightarrow x-y\in C$. Given a sequence of linear functionals $x^*_n:X\rightarrow\mathbb{R}$ ...

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538 views

### Completion of $C_0^{\infty}(\mathbb{R}^N)$ with norm $\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $

I have a question that I could not find it any where.
Is the completion of $C_0^{\infty}(\mathbb{R}^N)$ with the respect to norm
$$\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}...