# Questions tagged [fa.functional-analysis]

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

8,572
questions

2
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0
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36
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### Complex interpolation of Fourier-Lebesgue spaces and Lebesgue spaces

The complex interpolation of Lebesgue spaces $L^{p_0}$ and $L^{p_1}$ are well-known, that is,
$[L^{p_0}, L^{p_1}]_\theta = L^p$, where $(1-\theta)/p_0+1/p_1 = 1/p$ for some $\theta \in [0,1]$.
Now I ...

1
vote

0
answers

31
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### A convergence problem in the space of tempered distributions

Let $K(x):=|x|^{-\alpha}$ be a function on $\mathbb{R}^{n}\setminus\{0\}$ with $0<\alpha<n$. Suppose $\phi$ is a $C^{\infty}_{c}(\mathbb{R}^n)$ function such that
$$\text{(i)}\quad \text{supp}\...

1
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0
answers

53
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### Banach spaces in which every DP-set is a limited set

Let $X$ be a Banach space and $A\subseteq X$ be a bounded subset.
$A$ is a Dunford-Pettis set if every weakly null sequence $(f_n)$ in $X^*$ converges to $0$ uniformly on $A$, that is $$ \lim_{n\to\...

1
vote

1
answer

67
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### Is there an example of a causally supported Schwartz function on $\mathbb{R}^4$ invariant under the Lorentz transform?

I am working on $\mathbb{R}^4$ with the sign convention $(1,-1,-1,-1)$.
I wonder if there is Schwartz function $f(x)$ on $\mathbb{R}^4$ such that the support satisfies the condition $0<x^2 < 4m^...

1
vote

1
answer

55
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### Prove that $N_1(A_1,A_2)= N_2(A_1,A_2)$ when $A_1A_2=A_2A_1$ and $A_1$ and $A_2$ are normal operators

Let $\mathcal{L}(E)$ be the algebra of all bounded linear operators on a complex Hilbert space $E$.
On $\mathcal{L}(E)^2$, we have two equivalent norms:
\begin{eqnarray*}
N_1(A_1,A_2)
&=&\sup\...

1
vote

0
answers

61
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### Maximal domain of an unbounded linear operator in a weighted Hilbert-space

Let's consider the following (unbounded) linear operator. (So called transport operator in some context.)
$$ \mathrm{T}: \mathcal{H} \supset \mathcal{D}(\mathrm{T}) \to \mathcal{H} , f \mapsto \mathrm{...

4
votes

2
answers

123
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### Lebesgue differentiation theorem at boundary points for Sobolev traces

$\newcommand{\R}{\mathbb R}$
Let $\Omega\subset \R^d$ be a smooth, bounded open set and fix $p\geq 1$.
Fact 1: the usual Lebesgue differentiation theorem says that, if $u\in L^p(\Omega)$, then
$$
u(x)...

2
votes

1
answer

129
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### Open sets in the space of signed measures equipped with the Kantorovich–Rubinshtein norm

Let $X$ be a compact metric space and $\mathcal{M}(X)$ be the space of variational-bounded, signed Borel measures equipped with the Kantorovich–Rubinshtein norm, cf. [Section 8.3, 1]:
$$||\mu||_0:= \...

0
votes

0
answers

94
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### When is the image of $T \colon \ell^2 \to \ell^2$ a Gaussian random variable?

In finite dimensions, if $T$ is a linear operator and $x$ is a (centered) Gaussian random variable, then $Tx$ is again a (centered) Gaussian random variable.
Now suppose that $x$ is a (say, centered) ...

2
votes

0
answers

66
views

### Generalizations of elliptic chain complexes

I would like to know if it is possible to generalize the notion of elliptic chain complex of differential operators to different contexts, whether geometric or non geometric. I have in mind D-geometry....

1
vote

1
answer

33
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### Discrete uniqueness sets for the two-sided Laplace transform?

Let $f : \mathbb R_+ \to \mathbb C$ be a measurable and integrable function where $\mathbb R_+ = [0,\infty)$. The Laplace transform of $f$ is given by
$$
Lf(s) = \int_0^\infty f(x)e^{-sx} \, dx.
$$
A ...

1
vote

1
answer

95
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### Spectral theorem for unital $C^{*}$-algebras

Let $A$ be a unital $C^{*}$-algebra and $a \in A$ be normal, with spectrum $\sigma(a)$. Let $B = C^{*}(a)$ be the $C^{*}$-algebra generated by $1$ and $a$, which is abelian. Let $\hat{B}$ be the space ...

1
vote

1
answer

56
views

### The eigenvalues of the product $WD$ for some particular matrices

Let $D$ be a diagonal matrix in $M_{2n}(\mathbb{R})$ such that $D^2=I$ and Trace$(D)$=0
Suppose that $e_k$s are the standard vectors in $\mathbb{R}^{2n}$, that is
$$e_k=(0,\cdots 0,1,0,\cdots,0)^t$$...

0
votes

1
answer

138
views

### Looking for English version of a paper of Jean Ginibre

I am in serious need of an English translation for the following paper:
Séminaire Bourbaki by Jean Ginibre, Le problème de Cauchy pour des EDP semi-linéaires
périodiques en variables d’espace, d'après ...

11
votes

2
answers

434
views

### Existence of an open convex set

Let $T$ be a normed vector space, $K\subseteq T$ compact and convex and $O\subseteq K$ convex and open in $K$ (i.e. open w.r.t. the subspace topology of $K$ inherited by $T$).
Can we find an open set $...

1
vote

0
answers

140
views

### Weakly null sequences in projective tensor products

First, I'd like to record a question that may still be open. The snippet below is taken from DiestelPuglisi2009.
Second, let $E$ be a Banach space, $(u_n)$ be a weakly null sequence in the projective ...

0
votes

0
answers

59
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### Equicontinuity and estimating the modulus of continuity for linear operators on the Schwartz space

Let us consider a collection of continuous linear mappings on the Schwartz space $T_\alpha : \mathcal{S}(\mathbb{R^2}) \to \mathcal{S}(\mathbb{R^2})$ defined by
\begin{equation}
[T_\alpha(\phi)](x,y):=...

0
votes

0
answers

54
views

### Two questions about the vector-valued Lipschitz algebra

For a commutative Banach algebra $A$ and for any $0<\alpha<1$, let $\text{Lip}_\alpha(K,A)$ consist of all $A$-valued functions $f$ on a metric space $(K,\text d)$ with the property that $\rho_\...

1
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0
answers

106
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### Weak convergence using tensor product

I haven't got to see this argument used in the PhD thesis of [R. Ryan]: Applications of topological tensor products to infinite dimensional holomorphy, doctoral thesis, Trinity College, Dublin (1980), ...

6
votes

2
answers

343
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### Spectrum of operator involving ladder operators

The ladder operator in quantum mechanics are the operators
$$a^\dagger \ = \ \frac{1}{\sqrt{2}} \left(-\frac{d}{dq} + q\right)$$
and
$$a \ \ = \ \frac{1}{\sqrt{2}} \left(\ \ \ \!\frac{d}{dq} + q\...

0
votes

1
answer

66
views

### Minimal set of functions to characterize a distribution

In probability theory, there are a number of equivalent ways to characterize a distribution on $\mathbb R^n$. For example, the distribution of a random vector $X\in\mathbb R^n$ may be characterized by:...

-3
votes

0
answers

71
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### Isometric embeddings of two separable Banach spaces

It is well-known that a separable Banach space is isometrically embedded in $l^\infty$.
Consider $X\hookrightarrow Y$ two separable Banach spaces with a continuous embedding $i$. Therefore, there ...

1
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0
answers

35
views

### Dynamical formulation of the 2-Wasserstein distance for *discrete* matrix-valued measures

TL;DR: I want to find a definition generalizing "$t \mapsto \frac{1}{m} \sum_{k = 1}^{m} \delta_{x_k(t)}$ is a Wasserstein gradient flow" to matrix-valued probability measures.
Let $(X, d)$ ...

0
votes

1
answer

63
views

### The spectrum of the product $JA$ where $J=I_n\oplus (-I_n)$

Let $A$ be a real symmetric matrix in $M_{2n}(\mathbb{R})$with $A^2=I_{2n}$. Suppose that the Schur decomposition of $A$ is given by $A=\Lambda^t D \Lambda$. Let us consider the following matrix.
$$...

-1
votes

0
answers

28
views

### How to prove the convergence of the functions with the norm above the critical point of Kondrakov?

I have a sequence of smooth functions with norm 1 in $L_q$ space. I need to prove this sequence strongly converges to some function. But I lack the compactness theorem since, $q$ is above the critical ...

3
votes

0
answers

210
views

### Dunford-Pettis like properties for Banach spaces of operators

Let $E$ be a Banach space and $A\subseteq B(E)$ be a Banach subspace of operators on $E$.
Suppose $A$ satisfies the property (RCC) given below:
$$
\left.\begin{array}{l}
(x_n)\subseteq A \textrm{ ...

0
votes

1
answer

131
views

### What does "a universal tree" mean?

It is one of the concepts used in "ON THE REPRESENTATION OF CONTINUOUS FUNCTIONS OF SEVERAL VARIABLES AS SUPERPOSITIONS OF CONTINUOUS FUNCTIONS OF A SMALLER NUMBER OF VARIABLES", in the ...

3
votes

0
answers

86
views

### Recovering the matrix when the Schur decomposition of its blocks are known

Let E be a real symmetric matrix in $M_n(\mathbb{R})$ where $ n=2m$ and
$$E=\left(\begin{array}{cc}
G & X \\
X^t & H
\end{array}\right)$$
where $G,H,X$ are $m\times m$ matrices.
Suppose that $...

1
vote

0
answers

68
views

### Relationship between Beilinson’s resolution of the diagonal and functional analysis

I have been wondering for long enough to embarrass myself on here by asking: is there a reason why Beilinson’s resolution of the diagonal “Coherent Sheaves on Pn and Some Problems of Linear Algebra” ...

5
votes

1
answer

144
views

### States "absorbed" by a Haar idempotent on a compact quantum group

Firstly, a small question of nomenclature. If $(S,\bullet)$ is a magma, is there good terminology to relate $a$ to $b$ when
$$a\bullet b=b=b\bullet a?$$
Can we say that $b$ absorbs $a$? Can we say ...

0
votes

0
answers

79
views

### Finding set of best approximations from a point in $c_0$ to its subspace

Given $X$=$c_0$, null sequence space with sup norm. Consider a subspace $Y$ of $c_0$ consisting of elements of $c_0$ as, $Y=\{x\in c_0 : x_{2i}=i.x_{2i-1}, i \geq 1\}$. I need to find the set of best ...

2
votes

0
answers

79
views

### Reference for Schwartz kernel theorem on vector bundles

In this notes Linear Analysis on Manifolds by Chris Kottke at page 20 he has
Theorem $1.16$ (Schwartz kernel theorem, c.f. [Hör85] Thm. 5.2.1). Let $M$ and $N$ be a compact Riemannian manifolds with ...

1
vote

1
answer

183
views

### Sufficient condition for two norms to be equal

Let $\mathcal{L}(E)$ be the algebra of all bounded linear operators on a complex Hilbert space $E$.
On $\mathcal{L}(E)^2$, we have two equivalent norms:
\begin{eqnarray*}
N_1(A,B)
&=&\sup\left\...

2
votes

0
answers

99
views

### How to prove $\|I-P\| = \|P\|$ for any non-trivial projector? [duplicate]

I noticed that in the paper [1] this property is proved by explicitly computing the singular values of $P$ after realizing $P$ in finite dimension as oblique projection matrix. I am wondering if there ...

0
votes

0
answers

75
views

### Differentiation under the integral sign in higher dimensions [migrated]

Let's say I have a function $f(\mathbf{x},t)$, $f:\mathbb{R}^n\times\mathbb{R}\to\mathbb{R}^n$. Are there conditions under which the following holds:
$$
\frac{\partial}{\partial\mathbf{x}^T}\int_\...

0
votes

1
answer

144
views

### When some Fourier coefficients are fixed, can we control the extremals of the function?

Let $n$ be a odd number. Does there exist any $2\pi$-periodic continuous function $f
:\mathbb{R}\to \mathbb{R}$ such that the following points simultaneously hold?
1- $-n\lneqq f_{\min}$ (where $f_{\...

0
votes

1
answer

51
views

### Given positive $\epsilon$ and $c$, find a density $\phi$ such that $t\phi(\epsilon/t) \ge c \|\phi'\|_\infty$ for all positive $t$

A nice density (on $\mathbb R$) is function $\phi:\mathbb R \to \mathbb R$ such that
(1) $\phi(x) \ge 0$ for all $x \in \mathbb R$,
(2) $\int_{-\infty}^\infty \phi(x) \mathrm{d}x = 1$,
(3) $\phi$ is ...

3
votes

1
answer

349
views

### Boyd & Chua 1985: Is the proof of Lemma 2 correct?

$\newcommand\norm[1]{\lVert#1\rVert}\newcommand\abs[1]{\lvert#1\rvert}$I'm reading this article by Boyd and Chua [1], in which they prove the approximability of arbitrary time-invariant (TI) operators ...

2
votes

0
answers

49
views

### A division of real analytic functions

Problem statement
Let $f,g \in C^\omega(X,\mathbb{R})$ be two real analytic functions over a real Banach space $X$.
Assume that, for every $n \in \mathbb{N}$, there exists $C_n>0$ and $h_n \in C^\...

2
votes

1
answer

87
views

### Schauder bases in Banach spaces with a symmetric $k$-FDD

The Kalton-Peck Banach space $Z_2$ (see Section 6 in this paper) does not admit an unconditional basis, but it admits an unconditional, even symmetric, FDD (finite dimensional decomposition) into ...

0
votes

0
answers

90
views

### Solving a functional equation involving the exponential generating function of the Stirling numbers of the first kind

Let $F(x)=\sum_{n\geq 2m-1} f_n \frac{x^n}{n!}$, and $F_o=\sum_{n\geq m} f_{2n-1} \frac{x^{2n-1}}{(2n-1)!}$ for $m\geq 1$. Suppose
$$
F(x)+F(\frac{x}{1+x})=\frac{2[\ln (1+x)]^{2m}}{(2m)!}+\frac{2[\ln(...

-1
votes

0
answers

71
views

### Intersection of Hilbert spaces with Schauder basis plus a fixed singleton

Let $H$ be an infinite dimensional separable Hilbert space over $\mathbb{C}$.
Let $U \subset H$ be a Hilbert subspace of $H$.
Let $\{u_n\}_{n \in \mathbb{N}}$ be a Schauder basis for $U$.
Let $h \in H$...

3
votes

0
answers

152
views

### The essential norm where some Fourier coefficients are fixed

Let us denote $C_{2\pi}$ by the set of all $2\pi$-periodic continuous functions $f:\mathbb{R}\to \mathbb{R}$.
Q. Let $\phi\in C_{2\pi}$. Is the following statement valid?
$$\|\phi\|_2=\inf_{g\in C_{2\...

1
vote

0
answers

48
views

### The norm of Sobolev space involving the time

Question. Is the following way of writing the norm of a Sobolev space involving the time correct? I would be grateful for any help.
Let's assume we have a function
$$
\mathbf{u} (\mathbf{x}; t) = \...

1
vote

0
answers

134
views

### Could we characterize elements in the second dual by the character space?

Let $A$ and $B$ be two semisimple commutative Banach algebras. Assume that $A\mathbin{\tilde\otimes} B$ is a Banach algebra obtained by completing $A\otimes B$ with respect to a cross norm not ...

9
votes

1
answer

532
views

### Rigorous proof of the pentagon identity

I briefly recall the statement of the pentagon identity in quantum dilogarithm and cluster algebra.
For $b\in\mathbb{C}$ with $\operatorname{Re}(b)>0,\operatorname{Im}(b)\geq0$, Faddeev, Kashaev ...

1
vote

1
answer

50
views

### Show $A_0 + S$ is invertible, where $S$ is the Riesz projection of bounded $A_0$

Let $A_0$ be a bounded linear operator on a Hilbert space $H$. Suppose $0$ is an isolated point of the spectrum of $A_0$. Let $S$ be the corresponding Riesz projection, namely,
$$S = -\frac{1}{2\pi i} ...

1
vote

0
answers

121
views

### What pre knowledge does Mumford's Tata collections on theta need?

I am a sophomore and have the foundation of complex analysis and abstract algebra. I have learned Legendre elliptic integral and Jacobian elliptic function, and it is through this that I know theta ...

2
votes

0
answers

39
views

### When can convolutional integral operators be sampled

Consider an integral operator $F:C^{1/2}([0,T],\mathbb{R}^n)\rightarrow \mathbb{R}$ of the form
$$
f\mapsto \int_0^T f(t)^{\top}\kappa(T-t)dt,
$$
for some $\kappa\in C^{\infty}(\mathbb{R})(\mathbb{R},\...

1
vote

1
answer

189
views

### Norm continuity of the predual of a von Neumann algebra

Let $M$ be a von Neumann algebra and let $(p_i)$ be a net of projections in $M$ decreasing to $0$. Let $f\in M_{\ast} $, the predual of $M$.
It is well known that
$\| p_i f \|_{M_\ast}\to_{i} 0$
for ...