Questions tagged [fa.functional-analysis]

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

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Lipschitz map on positive definite cone of $n$-by-$n$ matrices

A function matrix $f : X \to \mathbb R$ is a convex Lipschitz continuous matrix function with Lipschitz constant $\mathrm L$ with respect to a fixed given norm $\|\cdot\|$, i.e., $|f(A)-f(B)| \leq \...
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Predual of $H^{\infty}(\mathbb{D})$

Is the predual of $H^{\infty}(\mathbb{D})$ contained in the maximal ideal space of $H^{\infty}(\mathbb{D})$?
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5 votes
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223 views

Existence of a limit of alpha-difference quotient for Hölder functions

Let $f:\mathbb{R}\to \mathbb{R}^d,d\geq 1,$ be an Hölder function with exponent $\alpha\in (0,1)$, meaning that \begin{equation} \sup_{x, y \in \mathbb R, \,x\neq y}\frac{|f(x)-f(y)|}{|x-y|^\alpha}<...
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Tracial linear functionals on an amenable Banach algebra

This post is related to an earlier question about Kazhdan property (T). The purpose of the snippet below is to briefly summarize the background for the question in this post. Question: Does there ...
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4 votes
1 answer
214 views

Weak* bounded and strong bounded are the same?

I have this problem at the moment which the strong topology $\beta (E;E^* )$ is defined, when $E$ is a locally convex space. This topology is generated by the basic open sets: $$U=\{x \in E : \sup_{f \...
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1 answer
80 views

Transforming two smooth densities to the same density

I am looking for an example of the following: Find a bijective, differentiable function $f$ and continuous probability density functions $q_1\ne q_2$ such that $f_*q_1=p=f_*q_2$, where $f_*$ is the ...
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28 views

Existence of Schauder basis incorporating a set of given linearly independent vectors

Assume $\mathcal{H}$ is a separable Hilbert space with orthonormal basis function $\{e_i\}_{i=1}^{\infty}$ that are countably infinite. We know there always exists a Schauder basis $\{ y_i \}_{i=1}^{\...
1 vote
1 answer
90 views

Inequality involving Sobolev spaces

Let us define $$ \mathbb{H}^{1} = H^{1}(-L,0) \times H^{1}(0,L) \ \ \text{and} \ \ \mathbb{L}^{2} = L^{2}(-L,0)\times L^{2}(0,L), $$ where $H^{1}(I) = \big\lbrace u \in L^{2}(I) \ \text{and} \ u_{x} \...
4 votes
1 answer
177 views

Johnson-Lindenstrauss over torus?

The Johnson-Lindenstrauss lemma is a famous result in dimensionality reduction — Given $m$ points in $\mathbb{R}^N$, and $\varepsilon >0$, there exists a quasi-isometry $f : \mathbb{R}^N\to\mathbb{...
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2 votes
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Does the snowflake $X^\alpha$ allows isometric embeddings into $L_1$ if $X$ does?

Question: Suppose that finite metric space $X$ allows isometric embedding to $L_1$. Does it mean that a snowflake space $X^\alpha$ allows isometric embedding to $L_1$ for every $0 < \alpha < 1$? ...
1 vote
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Multiple steps of the Gorelik principle

The following result is one of several non-linear Banach space theory results known as The Gorelik Principle. I am stating it here in a weaker form than what is in the literature, but the statement ...
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108 views

A measure on the group of homeomorphisms of $\mathbb T^2$

Let us consider the group of measure-preserving homeomorphisms of $\mathbb T^2$ (with transformations identified if they agree almost everywhere) called $G[\mathbb T^2, \mathcal L^2]$. We shall ...
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Schwarz space on the upper half-plane

Let us think of the Schwartz space $\mathcal{S}(\mathbb{R}^2_+)$ on the upper half-plane $\mathbb{R}^2_+=\mathbb{R}\times(0,+\infty)$ defined as $$ \mathcal{S}(\mathbb{R}^2_+)=\left\{f\in C^\infty(\...
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Does closure of exterior differential operator exist?

Let $\mathcal{L}^\rho_\nu$ be the set of all differential $\rho$-forms having $\nu$-th continuous dirivatives. Let $\mathcal{L}^\rho$ be the set of all $\rho$-forms with finite norm $\int |\phi|^2 &...
1 vote
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154 views

Building random homeomorphisms of the torus $\mathbb T^2$

In https://arxiv.org/abs/0912.3423, a family of random homeomorphisms of the circle is constructed. Main Question: Can the construction be generalized to higher space dimensions, e.g. to $\mathbb T^2$?...
1 vote
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83 views

Building random homeomorphisms of the circle

Given a positive Borel measure without atoms $\tau$ on the circle $\mathbb T =\mathbb R /\mathbb Z =[0,1)$ , in https://arxiv.org/abs/0912.3423 a homeomorphism $h:[0,1)\to [0,1)$ is defined as \...
3 votes
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On the relation between ellipticity and Fredholmness as properties of linear PDE's on Fréchet spaces of smooth sections

Let $M$ be a compact manifold equipped with finite rank vector bundles $E$ and $F$ with spaces of $C^{\infty}$ sections denoted $\Gamma(E)$ and $\Gamma(F)$ respectively. It is standard that a ...
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54 views

Density of Lipschitz functions in Bochner space with bounded support

Let $X$ and $Y$ be separable and reflexive Banach spaces with Schauder bases. Let $\mu$ be a non-zero finite Borel measure on $X$ and let $L^p(X,Y;\mu)$ denote the (Boehner) space of strongly p-...
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$|\partial $ as Fourier multiplier

I have the following nonlinear dispersive PDEs $$i \partial_t u- \partial_x^2 u =|\partial_x| |u|^2$$ where $f$ is some nice complex-valued function. I am trying to use the ansatz $u(t,x) = e^{i \...
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A basic question on analytic wave front set

Suppose $u$ is a smooth function on the closed unit disk centered at the origin in $\mathbb R^2$. Let us denote by $e_1$ and $e_2$ the unit vectors in the direction of $x$ and $y$ axis respectively. ...
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2 votes
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Malliavin-Shavgulidze (type) measures on the group of measure-preserving invertible maps on $\mathbb T$?

The Malliavin-Shavgulidze measures on $\operatorname{Diff}^{1}(I)$ (with $I$ an interval of $\mathbb R$) are defined as the image $W_{\sigma} \circ f^{-1}$ of the Wiener measure $W_{\sigma}$ with ...
3 votes
0 answers
88 views

Arithmetic progressions and removal lemmas for graphs in arithmetic combinatorics

As it is well known, one can gets a proof of Roth's Theorem concerning arithmetic progressions of length 3 (APs for short) by using the celebrated Ruzsa-Szemerédi triangle removal lemma for graphs. In ...
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2 votes
1 answer
78 views

Continuity of Legendre transform

Let $I \subset \mathbb{R}$ be an interval, and $f_n, f: I \rightarrow \mathbb{R}$ a convex function; then its Legendre transform is the function $f^{\ast}: I^{\ast} \rightarrow \mathbb{R}$ defined by $...
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6 votes
1 answer
187 views

Commutator ideal in nonunital C*-algebra

Let $A$ be a C*-algebra that has no one-dimensional irreducible representations, that is, there is no (closed) two-sided ideal $I\subseteq A$ such that $A/I\cong\mathbb{C}$. Let $J$ denote the (not ...
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4 votes
1 answer
95 views

"Open systems" version of Stone's Theorem for one-parameter groups of quantum operations

Let $H$ be a Hilbert space, which we interpret as a space of quantum states. If $U(t):H\to H$ is a unitary norm-continuous one-parameter group with $U(0)=I$, (essentially) Cauchy's functional ...
3 votes
1 answer
116 views

Proof that elements of Beppo-Levi-like spaces are functions (and not just distributions)?

Context. I am trying to undestand the theory underlying "Beppo-Levi"-like spaces defined as $$ H = \left\{f\in {\cal S}'(\mathbb{R}^d) \;\left| \; t\times\widetilde{f} \in {\cal L}^2(\mathbb{...
6 votes
1 answer
131 views

Sobolev space is spanned by distributions supported on half-lines?

I asked this question on Mathematics Stack Exchange previously. This seems to be a very basic property of Sobolev spaces, but I wasn't able to find a proof for it. For any $s \leq 1/2$, $$H^s(\mathbb{...
1 vote
1 answer
252 views

What's the name of this semi-group theorem?

I encountered this theorem, that for a bounded linear transform $L$ and a real parameter $t$ and initial data $u_0$, we have $$\frac{d}{dt} \exp(Lt)[u_0] = L \exp(Lt)[u_0].$$ What is the name of this ...
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To find a DFT for complex functions on a semigroup

For a certain commutative semigroup of integer size $n$, $G=(\{1,2,\dots,n\},\circ: x\circ y\mapsto \min(n,x+y))$, consider all complex functions on it, denoted by $\mathbb C[G]$ or $\mathbb CG$. ...
4 votes
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57 views

Characteristic of Sobolev space generated by Hörmander vector fields

Let $\Omega$ be an open bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial\Omega$. Suppose that $X=(X_{1},X_{2},\ldots,X_{m})$ are smooth vector fields defined on $\mathbb{R}^{n}$ and ...
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4 votes
2 answers
482 views

Unit ball of the sum space

Let $V$ be a vector space and $\|\cdot \|_1$ and $\|\cdot\|_2$ two norms on $V$. Let $\|\cdot\|_+$ be given by $$ \|v\|_+ := \inf_{v = v_1 + v_2} \|v_1\|_1 + \|v_2\|_2 $$ It is well-known that $\|\...
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1 vote
1 answer
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Is $N(A^n)=N^n(A)\;$ for a self-adjoint operator $A$?

A functional Hilbert space $\mathscr H=\mathscr H(\Omega)$ is a Hilbert space of complex valued functions on a (nonempty) set $\Omega$, which has the property that point evaluations are continuous i.e....
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1 vote
2 answers
272 views

Closed formula for this sum $\sum^\infty_{n=0}\frac{1}{n^4+n^2+1}.$ [closed]

How to calculate this sum $$\sum^\infty_{n=0}\frac{1}{n^4+n^2+1}.$$ Thank you in advance
4 votes
0 answers
82 views

Geometrical meaning of spaces that possess the weak* uniform Kadec-Klee property

What is the geometric meaning or interpretation of spaces that possess the weak* uniform Kadec-Klee property? I am writing the last part of my undergraduate thesis and I would like to add a comment in ...
6 votes
2 answers
344 views

Lipschitz mappings, covering dimension

Is there a compact metric space $X$ of covering dimension $2$ without a Lipschitz surjection on $[0,1]^2$? For a space $X$ with Hausdorff dimension greater than $2$, we have a negative answer (see ...
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1 answer
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Tensor product is complete?

Let $(V,\|\cdot\|_V)$ and $(W,\|\cdot\|_W)$ be Banach spaces and let the norm $\|\cdot\|_{V\otimes W}$ on the tensor product space $V\otimes W$ be admissible in the following sense: for $v\in V, w\in ...
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39 views

Semisimplicity of certain quotients of commutative Banach algebras

Let $A$ be a (Jacobson) semisimple regular commutative Banach algebra with a bounded approximate identity. For a given $f\in A^{\ast}$, let $k_f= \lbrace a\in A: Aa\subseteq\ker{f}\rbrace$. It is not ...
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Under what condition can I get the invertibility of the second order linear differential operator on compact manifold

Give a second order linear differential operator: $Lu=\Delta u + b^{i}u_{i} + c(x)u$ under what condition can I get the invertibility, clearly if $c(x) \le constant <0$ we can have the ...
7 votes
1 answer
166 views

How do people prove $\Gamma$-convergence in more complicated settings?

This is a soft question, I guess. $\Gamma$-convergence is a notion of convergence of functionals so that if $F_n$ $\Gamma$-converges to $F$, then cluster points of $\arg\inf F_n$ are minimizers of $F$....
1 vote
1 answer
112 views

Integration by parts for indicator of a sphere to indicator of a ball

Broadly speaking, I have a radial distribution on $\mathbb R^n$, i.e., the pdf only depends on the $\ell_2$-norm of the argument. I would like to obtain an expression for the pdf in the form $\int_{w=...
2 votes
1 answer
153 views

Bakry-Emery criterion

The most common use of the Bakry-Emery criterion is for the measure $\mu(x)=e^{-u(x)} /Z$ where $u \in \mathcal{C}^2$. I would like to ask for an application to a smaller class. Consider $u(x)=|x|^2 + ...
1 vote
0 answers
39 views

Holomorphic "quasi-interpolation" of a function sequence

I am interested in some sort of analytic interpolation. A toy version of my problem is as follows. Let $V \subset \mathbb{C}$ be a complex neighborhood of $[0,1]$. Assume there is some bounded ...
4 votes
0 answers
60 views

Multiplier algebra of Fock space

For any vector space, one may form the tensor algebra with multiplication being the tensor product. For a Hilbert space $\mathcal{H}$, the analogous construction is the Fock space $$ \mathcal{F}(\...
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2 votes
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Fourier transform harmonic oscillator eigenstates

The normalized eigenfunctions of the quantum harmonic oscillator are $$\psi_{n}(x)= \frac{1}{\sqrt{2^n n!}} e^{-x^2/2}H_n(x),$$ where $n \in \mathbb N_0$ and $H_n$ is the $n$-th Hermite polynomial, ...
4 votes
1 answer
288 views

Contact points for John's ellipsoid

Suppose $K$ is a centrally symmetric convex body in $\mathbb{R}^n$ and $E$ is the John's ellipsoid, the ellipsoid of maximal volume inside $K$. If $E$ and $K$ have exactly $2n$ contact points, say $(\...
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4 votes
1 answer
103 views

Whether a functional which preserves maximum for comonotone functions is monotone?

Let $X$ be a compactum (compact Hausdorff space). By $C(X,[0,1])$ we denote the space of continuous functions endowed with the sup-norm We also consider the natural lattice operations $\vee$ and $\...
4 votes
1 answer
116 views

A $C^*$ algebraic analogy of the concept of complemented subspace in the particular case of $\ell^\infty$

Let $A$ be a $C^*$ algebra. A $C^*$ subalgebra $C\subset A$ is said to be $C^*$ algebraic complemented of $A$ if there exist a $C^*$ subalgebra $D\subset A$ with $A=C\oplus D$ and the obvios mapping $...
1 vote
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47 views

$L^1$-valued Lipschitz extension problem on a simplex

Consider a regular $n$-simplex, and a map from the vertices to $L^1$. How can we find the minimum Lipschitz constant of an extension of this map to the entire simplex? Is there any literature or ...
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2 votes
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49 views

Semilinear elliptic equations in complex plane

Let $D$ denote the closed unit disk centered at the origin in the complex plane. Let $F: D \times \mathbb C \to \mathbb C$ be a smooth function. Is there any theory for well-posedness (in the sense of ...
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1 vote
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25 views

If $(\kappa_t)_{t\ge0}$ is a Markov semigroup with invariant measure $μ$, under which assumption is $t\mapsto\kappa_tf$ measurable for $f\in L^p(μ)$?

Let $(E,\mathcal E)$ be a measurable space; $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal E)$; $\mu$ be a finite measure on $(E,\mathcal E)$ which is subinvariant with respect to $(\...

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