# Questions tagged [fa.functional-analysis]

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

9,254
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### Lower bounds for truncated moments of Gaussian measures on Hilbert space

Let $\mu_C$ be a centered Gaussian probability Borel measure on a real separable Hilbert space $\mathcal{H}$ with covariance operator $C$. Denote the ball with radius $r$ in $\mathcal{H}$ centered at ...

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### Infinite tensor product of Hilbert spaces [duplicate]

Recently while reading an article I came across the usage of infinite tensor product of Hilbert spaces. I have got a basic understanding of doing computations in infinite tensor product while reading ...

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153
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### Reference request: Gaussian measures on duals of nuclear spaces

I am interested in constructive quantum field theory where Gaussian measures on duals of nuclear spaces (specifically, the space of tempered distribution $\mathcal{S}'(\mathbb{R}^n)$) play a key role. ...

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### Fréchet-valued symbols

Denote by $S^m \left ( \mathbb{R}^k \times \mathbb{R}^n \right)$ the usual space of symbols. Now let $E$ be a Fréchet Space. We can then define $S^m \left ( \mathbb{R}^k \times \mathbb{R}^n; E \right)$...

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### Definition of second quantization

The standard textbook for second quantization is Reed & Simon. However, I am a bit confused with their notation. They write:
Let $\mathscr{H}$ be a Hilbert space, $\mathcal{F}(\mathscr{H})$ the ...

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### How are coordinate charts constructed in noncommutative geometry?

In noncommutative geometry, one is given a triple $(A,D,H)$, where $A$ is a commutative C* algebra, $H$ is a Hilbert space, and $D$ is an operator. There is a somewhat long list of conditions that ...

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164
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### Large JN-sets in Banach spaces

For every infinite-dimensional Banach space $X$ there is a weak*-null sequence in the unit sphere of $X^\ast$. Does this extend under suitable circumstances to the non-separable setting?
Say that $X$ ...

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161
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### A continuous injection from the Hilbert cube to the real line?

Continuing an earlier "too good to be true" question that I posted recently, the same holds for the present question:
Is there a continuous injection from the Hilbert cube $[0,1]^{\Bbb N}$ ...

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### Quasimode construction on a compact Riemannian manifold

Let $M$ be a closed Riemannian manifold, $\Delta$ be the usual Laplace-Betrami operator on $M$ and $\gamma : [0, L] \to M$ be a stable elliptic periodic geodesic of length $L$. I have heard in several ...

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97
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### Solution to $a=e^t (t-r_1)(t-r_2)$ with Lambert $W$ function, where $r_1, r_2 $ are complex

Lambert $W$ works when $r_1$, and $r_2$ are real. However, I am trying to solve the equation when $r_1$, and $r_2$ are complex numbers.

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### Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space?

It seems too good to be possible, but:
Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space?
Here, the Cantor space $\{0,1\}^{\Bbb N}$ is equipped with the ...

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### Takesaki's duality in representation theory of $C^*$-algebras

In M.Takesaki's 1967 article titled A Duality in the Representation Theory of C-Algebras*, admissible operator fields are defined in order to generalize Gelfand transform to a non-abelian setting.
...

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64
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### Intersection of Sobolev Spaces

Suppose $\Omega$ is a bounded open subset of $\mathbb{R}^n$ with a "nice" boundary. We have the Sobolev spaces $W^{k,2}(\Omega)$, which are all contained within each other: $W^{m,2}(\Omega)\...

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### Best approximation rates of various classes of functions by truncated Fourier series

Let $f\in C([-1,1]^d)$ have periodic boundary, $N$ be a positive integer, and let $S_N(f)$ be the best approximation of $f$ by its truncated Fourier expansion truncated approximation
$$
S_N(f):=\sum_{...

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### Is it legitimate to use a delta function as a composite function? [migrated]

I am used to $\int f(x) \delta(x) \, dx = f(0) $ and the idea that this represents the action of a dual vector on $f(x)$ as an element of a vector space of functions, mapping it to the real number $...

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### Existence of a measurable maximizer

Let $F$ be a continuous cdf with full support on $[0,1].$ Let $A$ be a compact subset of $\mathbb{R}$ and $\mathcal{M}$ be the set of measurable functions $\alpha:[0,1]\rightarrow A.$ Let $\bar \alpha ...

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### Question about the Bessel operator

For $\nu>-1$ denote by $\{\lambda_{k,\nu}\}_{k\in\mathbb{N}}$ the succesive positive zeros of the Bessel function of the first kind $J_{\nu}$. The Bessel operator is given by
\begin{equation*}
L_\...

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### Reference request: log Sobolev inequality for uniform measure (uniform distribution over discrete set)

Suppose that $N \in \mathbb N_+$ is fixed and denote by $\mu = (\mu_0,\ldots,\mu_N)$ the uniform distribution on the set $\{0,1,\ldots,N\}$ (i.e., $\mu_n = \frac{1}{N+1}$ for each $0\leq n\leq N$). I ...

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### A question about associated operator on continuous functions space equiped with L2 norm

$$\text{For M a connected compact manifold, T is in }C^{1+\nu}(M,M) \text{ with } \nu\in(0,1),\\ \text{i.e. DT is some Hölder continuous function with Hölder exponent }\\ \text{, Denote m as the ...

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### Is such a Banach space $X$ isometrically isomorphic to a Hilbert space or not?

Let $X$ be a real or complex Banach space. $X$ satisfies:There exists real or complex series $\{a_k\}_{k=1}^n,\{b_k\}_{k=1}^n$ (which satisfies that:$\begin{cases}
a_k,b_k\in \mathbb{R}\ \forall k=1,\...

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### Dependence of functional integral on the function space

In physics, the following functional integral is considered
\begin{gather}
Z[J]= \int Df \exp(-\int d^dx( f\Box f+\lambda f^4 +Jf ))
\end{gather}
It is usually said that the integration is performed ...

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293
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### Necessary conditions for convergence of convolution

In math.SE, I've asked a question about the convergence of convolution of two functions which have bilateral Laplace transform and also have disjoint Region Of Convergence (ROC) but the question didn'...

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### Series representation of functions

Let $H$ be a Hilbert space, consisting of functions $f:\mathbb{R} \to \mathbb{R}$. Let
$$
V = \left\{ f_J \in H: f_J= \sum_{j=1}^J c_j^{(J)} g_j, c_j^{(J)}\in\mathbb R, J\in \mathbb N \right\}
$$
...

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### Array-determined operator ideals

For a Banach space $X$, we, of course, know what it means for a sequence to be weakly null (to converge to zero in the weak topology).
An array in the Banach space $X$ is a sequence of sequences, $(...

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137
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### Topology of ${\mathcal D}(\Omega)$ (space of test functions)

I have seen two approaches to the topology of ${\mathcal D}(\Omega)$:
(i) Let $K$ be a compact subset of $\Omega$ and consider the subset ${\mathcal D}_K(\Omega)$ of test functions with support ...

3
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467
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### A strange condition of convexity?

During my research, I come across this question.
Let $f \in C^2(\mathbb R, \mathbb R_+^*)$ with $\forall x \in\mathbb R, f'(x) \geq |f''(x)+f(x)|$.
Is it true that $\forall x \in \mathbb R, f''(x) \...

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### Existence of the limit of periodic measures

Let $T: X \to X$ be a continuous map over a compact metric space. We say that a measure $\mu$ is $T$-invariant if $T_{\ast} \mu= \mu$. We denote by $M(X, T)$ the space of all $T$-invariant Borel ...

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134
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### Prove if the fractional Laplacian of a function is bounded

Take $s\in (0, 1)$. I am trying to understand if $(-\Delta)^s (\log(1+x^2))$ is bounded, that is if there exists $R>0$ such that $|(-\Delta)^s (\log(1+x^2))|\le R$.
Here $(-\Delta)^s$ is the ...

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39
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### Instances of c-concavity outside of optimal transport?

Let $X$ and $Y$ be metric spaces, and let $c:X\times Y\rightarrow \mathbb{R}$ be a nonnegative function which we refer to as a cost. For any $\phi:X\rightarrow \mathbb{R}$ and $\psi:Y\rightarrow \...

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### What are alternative or equivalent definitions of a positive-definite function on a group?

The standard definition of a positive-definite function on a group goes as follows:
Let $\varphi : G \rightarrow L(H)$, where $G$ is a group (with an involution) and $H$ a Hilbert space. $L(H)$ is the ...

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### If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dotsb+ B_n'$ also satisfies the same inequality

Related: On a deceptively tricky calculus problem.
The way that Leonard Gross proves the log Sobolev inequality is in the following stages:
He proves that for any operator $B$ that satisfies the log ...

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### Conditions for completely positive maps to act homomorphically across multiple subalgebras

For a completely positive (CP) map $u: A \to A'$ of $C^*$-algebras $A, A'$, the concept of multiplicative domains characterizes the largest subalgebra of $A$ on which $u$ behaves as a $*$-homomorphism....

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### Topology of an orbit space constructed from a Fréchet space under the "local" action of some "smooth" group

Let $G$ be a nontrivial connected compact subgroup of the general linear group $\operatorname{GL}(\mathbb{R}^3)$. For example, we may take $G$ to be $\operatorname{SO}(3)$.
Next, let $\mathcal{S}(\...

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146
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### Equivalent definition for Skorokhod metric

I have a question about the Skorokod distance on the space $\mathcal{D}([0,1],\mathbb{R})$:
$$
d(X,Y):= \inf_{\lambda \in \Lambda}\left( \sup_{t\in [0,1]}|t-\lambda(t)|\vee \sup_{t\in [0,1]}|X(t)-Y(\...

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### Upper bound Hölder norm of the solution to the non-linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (u (t, x))|^2 u(t, x) \}$

We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb R \to \mathbb R$ belong to the Hölder space $C^{1, \alpha}_b (\mathbb R)$ for some $\alpha \in (0, 1)$. Let $u : \mathbb T \times \...

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1
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### An integrable estimate of the Hölder constant of the map $x \mapsto \int_{\mathbb R^d} f(y) \partial_1 \partial_1 g_t (x-y) \, \mathrm d y$

Let $(g_t)_{t>0}$ be the Gaussian heat kernel on $\mathbb R^d$, i.e.,
$$
g_t (x) := (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}},
\quad t>0, x \in \mathbb R^d.
$$
Let $f : \mathbb R^d \to \...

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### Preimage of null sets under a monotone increasing function

Let $I\subseteq \mathbb{R}$ be a closed bounded interval and $f:I \to I$ a monotonic increasing function and $S$ the countable set of points $s$ such that $|f^{-1}(s)| > 1$. Is the following ...

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### The operator $D^{p}\colon \mathcal{S}\subset L^{1}(\gamma)\to L^{1}(\gamma)$ is closable for every integer $p =1,2,\dots$

I am reading Nourdin and Peccati’s textbook (Normal Approximations with Malliavin Calculus From Stein’s Method to Universality). My question is about Lemma 1.1.6. Which says
Lemma 1.1.6:
The operator $...

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154
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### Equivalent Littlewood-Paley-type decompositions

The theory of Besov and Triebel-Lizorkin spaces usually proceeds by taking a dyadic decomposition of unity, i.e. some non-negative functions $\psi_0,\psi \in C_c^\infty(\mathbb{R})$ such that
\begin{...

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87
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### Construct next polynomial from predecessor and resulting GCD

I have a sequence of polynomials built from an interpolation derived in a combinatorial problem. For each integer value of a parameter $n$ there is a different polynomial.
After trying to find a way ...

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2
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501
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### Prove that the following function is positive

Consider the following function:
$$K(x, y; t) = \sum_{n \geq 0} \frac{e^{-(2n+1)t}}{\sqrt{\pi} 2^n n!} H_n(x) H_n(y) \exp\left(-\frac{(x^2 + y^2)}{2}\right)
$$
This is Mehler's kernel, and can be ...

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153
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### Sobolev regularity via Laplace spectrum

Fix a positive integer $n$ and let $\mu$ be the uniform measure on the sphere $\mathbb{S}^n$, with respect to its usual Riemannian metric $g$. Let $\nabla$ be the Laplacian on $(\mathbb{S}^n,g)$ and ...

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### In what sense does the Laplacian on compact intervals converge to one on all of $\mathbb{R}$?

I guess this topic may have been addressed somewhere but I cannot really find a reference myself, so I ask here.
For each $N \in \mathbb{N}$, consider the Laplacian $\Delta$ on the interval $[-N,N]$ ...

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150
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### Generalization of polynomial coefficients

I'm dealing with a hard combinatorial problem where for every positive integer value of a variable $n$ I have to calculate a list of numbers, specifically $n^2$, that depend on $n$ and its list index ...

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### Question about the ''crater'' in mountain-pass theorem while reading a paper of solving mean-field equation by mountain-pass theorem

Actually, I'm reading a paper which finds the saddle point of a functional, of course the unbounded below energy functional will suggest a potential saddle, but the structure of mountain pass is the ...

2
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36
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### Functional of fully nonlinear equations

Let $\left(\mathcal{M}, g_0\right)$ be a compact Riemannian manifold of dimension $n>2$ and denote by 'Ric' and $R$ respectively the Ricci tensor and the scalar curvature. The $k$-Yamabe problem is ...

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115
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### Gelfand's representation on matrices: construct maximal ideal in matrix algebra

I would like to see a constructive proof (some algorithm?) of the following statement:
Let $A_1, A_2, \dotsc ,A_k \in M_n(\mathbb C)$ be some commuting matrices, let $B$ be the commutative algebra (...

2
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1
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144
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### Growth rate of elementary sequences

We consider three sequences $(x_n),(y_n),(z_n)$, where $(x_n) \in \ell^1$ is positive and the other two sequences are merely assumed to be positive, i.e. $y_n,z_n \ge 0$ where $0<z_n<z_{n+1}$ is ...

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63
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### Is a bounded measurable convex function above its interior lower semi-continuous convex envelope?

Let $E$ be a locally convex topological vector space, let $C$ be a convex set which matches the closure of its relative interior $\mathring C=\{ x\in C : \forall y\in C,\exists z\in C,~x\in\mathopen]y,...

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34
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### Finding an element of Gelfand triple with a designated time derivative

Let $V$ be a real separable Banach space and $H$ be a real separable Hilbert space such that
\begin{equation}
V \subset H \subset V'
\end{equation}
where $V'$ is the dual of $V$ and the inclusions are ...