Questions tagged [fa.functional-analysis]

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

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Where does the Laplace transform come from?

The Gelfand transform on the commutative Banach *-algebra $L^1(\mathbb{R})$ is just the Fourier transform. Q. What can we say concerning the Laplace transform?
ABB's user avatar
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Is there a 'certainty' principle?

Heisenberg's uncertainty principle is a restriction on which probability distributions can describe the position and momentum of a quantum particle. In mathematical terms it says that if $\psi\in L^2$ ...
Oscar Cunningham's user avatar
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Extension of positive functionals II

This is a follow-up to Extension of positive functionals. Assume that $X=R^n$ with the canonical order (I indicate with $K$ the positive cone, $x \in K$ iff $x_i \ge 0$ for all $i$) and let $L:M \to R$...
Giorgio Metafune's user avatar
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1 answer
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Question regarding the Wick tensor in white noise analysis

I have a question regarding the definition of Wick tensor in the framework of the white noise analysis. To put some context to the question we start with the following Gel'fand triple $$S(\mathbb R)\...
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Representation theorem for quadratic form on Hilbert space

I think my question is more suitable for Mathematics Stack Exchange than to MathOverflow but I've already posted two related questions there and I got even more confused, so maybe I can clarify things ...
MathMath's user avatar
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14 votes
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strong topologies on $C_c^\infty$

UPDATE (27/08/2020): I realized after a comment from Jochen Wengenroth that there was at least one false premise behind my question, owing to the fact that analysts sometimes use the words "...
Chris Wendl's user avatar
8 votes
1 answer
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The double dual of the unitization of a $C^*$-algebra

I am studying the proof that if $A$ is a $C^*$-algebra such that $A^{**}$ is a semidiscrete vN algebra, then $A$ has the completely positive approximation property (CPAP). I was able to handle the ...
Just dropped in's user avatar
1 vote
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Property $(\mathcal{L}(\phi),\phi)\geq 0$ about a operator $\mathcal{L}$

Consider the operator $\mathcal{L} : H^2(\mathbb{T}_L) \subset L^2(\mathbb{T}_L) \longrightarrow L^2(\mathbb{T}_L)$ given by $$\mathcal{L} = -\omega \partial_x^2+3\varphi^2-1,$$ that is $$\mathcal{L}(...
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Extension of positive functionals

Let $X$ be a function space as $C(K)$ or $L^p$, with its usual norm and order, that is $f \le g$ if and only if $f(x) \le g(x)$ for a.e. $x$. If $M$ is a subspace of $X$ and $L:M \to \bf R$ is a ...
Giorgio Metafune's user avatar
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Generalized convexity

Let $X$ be a vector space. The positive-homogeneous function $\|\cdot\|$ is said to be a quasinorm if $\|x+y\|\le K(\|x\|+\|y\|)$, for some $K\ge1$; it is a norm if $K=1$. Question: 1. (terminology) ...
Aryeh Kontorovich's user avatar
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Estimate involving Besov norm

When reading some old notes of my advisor on interpolation spaces, I bumped into a problem I can't quite wrap my head around. Here are the details. For $p\in(0,\infty)$ a $p$-variation semi-norm of a ...
Tony419's user avatar
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Consider a net of weak order units in a Riesz space converging in order to a weak order unit. Is there a tail whose infimum is a weak order unit?

Let $X$ be an extremally disconnected (the closure of an open set is open) compact Hausdorff space, and consider the Riesz space $C^\infty(X)$ of continuous functions from $X$ to the extended real ...
Mark Roelands's user avatar
8 votes
2 answers
524 views

Are (completely) positive maps approximated by normal (completely) positive maps?

Let $\mathcal{H}$ denote a Hilbert space and $B(\mathcal{H})$ denote the algebra of all bounded operators on $\mathcal{H}$. By recognizing the (Banach) dual of $B(\mathcal{H})$ with the double dual of ...
Manish Kumar's user avatar
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134 views

Characterizing Besov spaces in terms of p-variation

For $s>1/p$ the Besov space $B_{p,q}^s([0,1])$ can be characterized in terms of the $p$-variation: Let $p,q \in (1,\infty)$ and $s \in (0,1)$, $s>1/p$. A function $f:[0,1] \to \mathbb{R}$ is in ...
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Fréchet derivative of evaluation-like functional (multivariate)

I'm fairly new to functional calculus but and posting here since the question seems more appropriate than for MSE. When coming across this post I could not help but wonder the following. Let $H$ be ...
ABIM's user avatar
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The conformal map from interior of ellipse to interior of the unit disk (property check)

Based on example 5 (page 546) and example 7 (page 550) of the book "Applied and computational complex analysis. Volume 3, Wiley, 1986" written by Peter Henrici, if $a,b>0$ satisfies $a^2-...
Fei Cao's user avatar
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3 votes
1 answer
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Existence of probability measure on the circle with given Fourier coefficients

We say that a Hermitian symmetric (i.e., $f_{-n} = f_n^*$ for any $n \in \mathbb{Z})$ sequence $(f_n)_{n\in \mathbb{Z}}$ is positive-definite if, for any $N \geq 0$ and any $z_0 , \ldots, z_N \in \...
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Conditions on triangle inequality for integral kernel

Consider $\int_RK(x,y)f(y)dy$, where $K(x,y) \in M_+(R^2)$. Let $L(t,s)$ be an iterated rearrangement of $K$. Let also $$ A(t,v)=\int_0^{1/v}L(1/t,s)ds, $$ which is decreasing with $v$ and ...
user124297's user avatar
4 votes
3 answers
241 views

A functional integral inequality

Suppose $f:I=(0,1)\to \mathbb R$ is a continuous function that satisfies $$ \int_I f(t) e^{at}\,dt \geq 0\quad \text{for all $a \in \mathbb R$}.$$ Does it follow that $f\geq 0$ on $I$?
Ali's user avatar
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16 votes
1 answer
506 views

Balls in Hilbert space

I recently noticed an interesting fact which leads to a perhaps difficult question. If $n$ is a natural number, let $k_n$ be the smallest number $k$ such that an open ball of radius $k$ in a real ...
Bruce Blackadar's user avatar
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1 answer
183 views

a question about vector valued Banach spaces

I wonder the difference between $L^1(\mu\times\nu)$ and $L^1(\mu;L^1(\nu))$, as if partial derivatives can be exchanged with integration in the second spaces in many articles. In Folland's real ...
WPJ's user avatar
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Sums over products over short paths in an expander graph

Let $\Gamma=(V,E)$ be an undirected graph of degree $d$. (Say $d$ is a large constant and the number of vertices $n=|V|$ is much larger.) Let $W_0$ be the space of functions $f:V\to \mathbb{C}$ with ...
H A Helfgott's user avatar
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3 votes
2 answers
556 views

When is the periodisation of a function continuous?

Consider a function $f\in\mathcal{C}_0(\mathbb{R})$, where $\mathcal{C}_0(\mathbb{R})$ denotes the space of bounded continuous functions vanishing at infinity. I am interested in the $T$-periodisation ...
MatthieuMeo's user avatar
14 votes
2 answers
868 views

Do distance functionals separate probability measures?

Let $(\Omega,d)$ be a compact metric space and $\mathcal P(\Omega)$ its space of Borel probability measures. Let $D=\{ d_p\mid p\in\Omega\}$ where $d_p(x)=d(p,x)$ be the set of all "distance ...
Christian Bueno's user avatar
3 votes
0 answers
250 views

Eigenvalue bounds and triple (and quadruple, etc.) products

Very basic and somewhat open-ended question: Let $A$ be a symmetric operator on functions $f:X\to \mathbb{R}$, where $X$ is a finite set. Assume that the $L^2\to L^2$ norm of $A$ is $\leq \epsilon$, i....
H A Helfgott's user avatar
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What is the weak limit of $f_n \ \mathrm{sign}(f_n - 1)$ if $f_n \to f$ weakly in $L^p([0,1])$?

Let $f_n: [0,1] \to \mathbb R$ be a uniformly bounded sequence in $L^p$. Then there exists a subsequence such that $f_{n_k} \to f$ weakly in $L^p([0,1])$. What is the weak limit of the sequence of ...
Lao's user avatar
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0 answers
285 views

Frêchet differentiability of the composition on a suitable Banach space

Let $E$ be a real Banach space included in the space of functions $C^1$ of $\mathbb R\to \mathbb R $ and and $T:E\to E$ defined by $T(f)=f\circ f$. I am looking for an example of space E such that T ...
Pascal's user avatar
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5 votes
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264 views

Riesz Representation Theorem for $L^2(\mathbb{R}) \oplus L^2(\mathbb{T})$?

The spaces $L^2(\mathbb{R})$ (square-integrable functions) and $L^2(\mathbb{T})$ (1-periodic square-integrable functions, considered over the real line $\mathbb{R}$) are two subspaces of the space of ...
Goulifet's user avatar
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6 votes
2 answers
490 views

A question on Grothendieck space

A Banach space $X$ is said to be Grothendieck if the weak and the weak* convergence of sequences in $X^{*}$ coincide. I have the following two questions. Question 1. A Banach space $X$ is Grothendieck ...
Dongyang Chen's user avatar
2 votes
0 answers
41 views

Analysis of coefficients for quickly vanishing analytic vector field

Let $u = (u_1, u_2, u_3): \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a divergence-free analytic vector field for $n =3$ or $n =4$, i.e., $u_i : \mathbb{R}^n \rightarrow \mathbb{R}$ are analytic ...
tobias's user avatar
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0 answers
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Double commutant theorem when $C^*$-subalgebra does not contain identity operator $1$

Double commutant theorem: For a unital $C^*$-subalgebra $M \subset B(H)$, one has $$\smash{\overline M}^\text{SOT}=\smash{\overline M}^\text{WOT}=M''.$$ My question: For a $C^*$-subalgebra $M \subset ...
user62498's user avatar
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5 votes
1 answer
149 views

$C^j$-topology considered by Greene and Krantz

My question is about the $C^j$ topology used by Greene and Krantz in their paper "Deformations of Complex Structures, Estimates for the $\bar{\partial}$-equation, and stability of the Bergman ...
Pita's user avatar
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10 votes
1 answer
509 views

discontinuous functions on the Sobolev borderline

The Sobolev embedding theorem implies that every function of class $W^{k,p}$ on a reasonable $n$-dimensional domain is continuous if $kp > n$. Cases with $kp=n$ are known as "borderline" ...
Chris Wendl's user avatar
5 votes
2 answers
220 views

Analytic approximations of smooth vector fields

Let $M$ be the set of smooth divergence-free vector fields $u$ on $\mathbb{R}^3$ with $$|\partial_x^{\alpha} u(x)| \leq C_{\alpha K}(1+|x|)^{-K}$$ on $\mathbb{R}^3$ for any $\alpha,K$. Further, we ...
tobias's user avatar
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6 votes
1 answer
608 views

Closed convex hull in infinite dimensions vs. continuous convex combinations

tl;dr: When is the closed convex hull of a set $K$ equal to the set of "continuous" convex combinations of $K$? I am essentially asking for the most general, infinite-dimensional analogue of ...
user163625's user avatar
5 votes
1 answer
145 views

Why is density and separability needed for uniqueness of weak (time) derivatives?

Let $X,Y$ be Banach spaces with $X \subset Y$. Recall that $u \in L^1(0,T;X)$ has weak derivative $g \in L^1(0,T;Y)$ if $$\int_0^T u(t)\phi'(t) = -\int_0^T g(t)\phi(t) \qquad\forall \phi \in C_c^\...
StopUsingFacebook's user avatar
2 votes
1 answer
68 views

Equicontinuity-like property of a convex compact set

Let $X$ be a Tychonoff topological space and let $x\in X$. Let $B\subset C(X)$ be convex and compact in the topology of pointwise convergence, and such that $f(x)=1$, for every $f\in B$. Is there an ...
erz's user avatar
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0 votes
0 answers
45 views

Critical Growth of Dimension for Dense Cover by Linear Subspaces

Let $X$ be a separable Banach space of dimensional $>2$. When does there exist a sequence positive integers $\{N_n\}_{n \in \mathbb{n}}$ such that For any sequence of distinct finite-dimensional ...
ABIM's user avatar
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1 vote
0 answers
24 views

On some bounds on two constants concerning the disconnectedness of the spectra of small perturbations of operators

Let $H$ be a separable, infinite dimensional, complex Hilbert space. In the book: Jiang, C. L.; Wang, Z. Y. (1998). Strongly Irreducible Operators on Hilbert Space. CRC press above the statement of ...
Manuel Norman's user avatar
5 votes
1 answer
162 views

Critical Smoothness on Besov Spaces $B^s_{p}$: how does it evolved with $p$?

We denote by $B_{p}^s(\mathbb{T}) := B_{p,p}^s(\mathbb{T})$ the Besov space over the circle $\mathbb{T}$ with parameters $p=q \in (0, \infty]$ and smoothness $s \in \mathbb{R}$. For $p>0$ fixed and ...
Goulifet's user avatar
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4 votes
0 answers
132 views

Smooth dependence of solution to elliptic pde depending on parameter

I have a question in mind but let me generalize it slightly. Suppose I am looking at some pde like $$-\Delta u + t u = f(u)$$ in $B_1$ (here $u=u(x)$) with $u=0$ on $ \partial B_1$ where $B_1$ is ...
Math604's user avatar
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1 vote
1 answer
313 views

$L^p$ compactness for a sequence of functions from compactness of product with cut-off

Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
Zac's user avatar
  • 161
4 votes
1 answer
963 views

Functional derivative of differential entropy

I have trouble finding the derivative of the differential entropy w.r.t the probability density function, i.e. what is $\frac{\delta F[p]}{\delta p(x)}$, where $F[p] = \int_X p(x)\ln(p(x))dx$, and $p(...
Jan Rathjens's user avatar
4 votes
1 answer
192 views

A kind of holomorphicity of maps on Hilbert space

Let $H$ be an infinite dimensional seperable Hilbert space. Is there an Irreducible involutive sub algebra $D$ of $B(H)$ with the following properties?: 1)For every open set $U\subset H$ and every ...
Ali Taghavi's user avatar
2 votes
0 answers
70 views

Example of BV vector field $c$ without bounded divergence such that $u$ is bounded where $u_t + div(cu) = 0$

What is an example of vector field $c: \mathbb R_+ \times \mathbb R^N \to \mathbb R^N$ with $c \in L^1(\mathbb{R}_+, BV(\mathbb R^N))$ without bounded divergence $div_x c$ but such that there exists a ...
Riku's user avatar
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3 votes
0 answers
72 views

A holomorphic shrinking of a domain into a compact subset

This question is related to these two. Let $X\subset \mathbb{C}^{n}$ be a bounded domain. I am interested in the following property: there is a sequence of continuous maps $\varphi_n:\overline{X}\to X$...
erz's user avatar
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2 votes
0 answers
60 views

Uniqueness of solution to Cauchy problem with quadratic nonlinearity

Consider the non-linear differential operator $$\mathfrak{L}: \ C^2((0,T)\times\mathbb{R}^2)\ni\varphi\equiv\varphi(t,x,y) \, \mapsto \, \partial_x^2\varphi + (\partial_x\varphi)^2.$$ For $U\subset\...
cts12's user avatar
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2 votes
0 answers
62 views

Integral convergence with two sequences of functions

I came across this theorem just stated but has not proved and marked by 'it is easy to see'. Theorem If $u_m$ and $v_m$ converges to $u$ and $v$ in $L^2([0,T];H^1(\Omega))$ weakly and $L^2([0,T];L^2(\...
Lev Bahn's user avatar
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2 votes
0 answers
114 views

Is $A$ is small on bounded functions, is there a large subdomain on which $A$ is small?

Let $A$ be a symmetric linear operator of norm $\leq 1$ on the space of functions $f:S\to \mathbb{R}$, where $S$ is a set with $N$ elements. Define the inner product $\langle \cdot,\cdot\rangle$ by ...
H A Helfgott's user avatar
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1 vote
1 answer
227 views

Does the Skorokhod space with the uniform topology admit a smooth partition of unity?

Does the Skorokhod Banach space $D[0,1]$ (cadlag functions equipped with the uniform norm) admit a smooth partition of unity? I found Johanis - Smooth partitions of unity on Banach spaces, which ...
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