# Questions tagged [fa.functional-analysis]

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

6,739
questions

**4**

votes

**0**answers

46 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 ...

**0**

votes

**0**answers

46 views

### Continuity of operator exponential on subspace

Let $H$ be a separable Hilbert space with overcomplete basis $(e_t)_{t \in (0,\infty)}$ such that $\langle e_t, e_s \rangle = e^{-\vert t-s \vert}.$
We can then define the projection $P_t$ which is ...

**1**

vote

**1**answer

122 views

### 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 ...

**1**

vote

**0**answers

53 views

### 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-...

**3**

votes

**1**answer

170 views

### 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 \...

**1**

vote

**0**answers

59 views

### 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 ...

**4**

votes

**3**answers

189 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 $I$?

**0**

votes

**0**answers

23 views

### parametric model with parameters described as gaussian processes

Let's assume that I have some data $y_{t_i}$ (i = 0, 1, ..., N) and a model $\hat{y}(a(t), b(t))$, where the parameters of my model (a, b) evolve with time t in a stochastic manner. I am wondering if ...

**0**

votes

**0**answers

50 views

### Generalization of Banach-Lamperti theorem to element-wise nonlinear transformations

According to the Banach-Lamperti theorem, every linear isometry $T$ of $\ell_p = \ell_p(\mathbb{N})$ (with $1 \leq p < \infty,𝑝\ne 2$) is of the form $T:(a_n)↦(\epsilon(n)a_{\sigma(n)})$, where $\...

**16**

votes

**1**answer

270 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 ...

**0**

votes

**1**answer

124 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 ...

**3**

votes

**1**answer

194 views

### 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 ...

**2**

votes

**2**answers

143 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 ...

**12**

votes

**2**answers

680 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 ...

**3**

votes

**0**answers

209 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....

**2**

votes

**0**answers

76 views

### 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 ...

**4**

votes

**0**answers

237 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 ...

**5**

votes

**0**answers

182 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 ...

**5**

votes

**1**answer

213 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 ...

**2**

votes

**0**answers

39 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 ...

**0**

votes

**0**answers

46 views

### 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 ...

**5**

votes

**1**answer

97 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 ...

**9**

votes

**1**answer

165 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" ...

**5**

votes

**2**answers

154 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 ...

**5**

votes

**1**answer

241 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 ...

**5**

votes

**1**answer

102 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^\...

**2**

votes

**1**answer

58 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 ...

**0**

votes

**0**answers

22 views

### Spectral gap of continuous-time Markov chain on nonnegative integers: The geometric long indel length chain

Let $S=\mathbb Z_{\ge0}$ be the nonnegative integers, and let $T=\mathbb R_{\ge0}$ be the nonnegative real numbers. Next, let $\gamma\in(0,1),r\in(0,1),$ and let $Q=(Q_{n,m})_{n,m\in S}$ be such that ...

**3**

votes

**0**answers

42 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 ...

**1**

vote

**0**answers

18 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 ...

**5**

votes

**1**answer

98 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 ...

**3**

votes

**0**answers

56 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 ...

**2**

votes

**1**answer

173 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(...

**2**

votes

**1**answer

116 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(...

**3**

votes

**0**answers

73 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 ...

**2**

votes

**0**answers

45 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 ...

**0**

votes

**0**answers

65 views

### what is $\{N_\mu\oplus N_\mu\}'$, where $N_\mu f=zf$ for each $f\in L^2(\mu)$

Definition: The commutant $\mathcal{S}'$ of a subset $\mathcal{S} \subset B(H)$ is $ \{A \in B(H) : AB=BA \ , \ \forall B \in \mathcal{S} \} $. Let $H$ is Hilbert space and $μ$ be a regular Borel ...

**3**

votes

**0**answers

67 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 $\varphi_n:\overline{X}\to X$...

**2**

votes

**0**answers

56 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\...

**2**

votes

**0**answers

46 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(\...

**2**

votes

**0**answers

107 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 ...

**0**

votes

**1**answer

101 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 ...

**1**

vote

**0**answers

71 views

### Eigenvalues of product of operators

Let $A,B$ be two Trace class operators with spectral decomposition $\sum_{j\geq 1} \lambda_j \phi_j(\cdot)\otimes \phi_j(\cdot)$ and $\sum_{j\geq 1} \gamma_j \psi_j(\cdot)\otimes \psi_j(\cdot)$ ...

**5**

votes

**3**answers

314 views

### Noncommutative torus as a von Neumann algebra

Le $\theta$ be irrational. One can define the noncommutative torus $A_{\theta}$ as a universal algebra generated by two unitaries $u,v$ satisfying the relation $vu=e^{2 \pi i \theta} uv$. This is an ...

**5**

votes

**1**answer

224 views

### improved Sobolev embedding

This is probably not a research level question but I am struggling with the geometry. My question is related to whether some monotonicity can increase the range of exponents in the Sobolev embedding.
...

**1**

vote

**1**answer

147 views

### A question of uniqueness

Let $u$ an harmonique function on $\Omega=(a,b)\times (0,+\infty)$ and boundary conditions :
$\displaystyle u(a,y)=u(b,y)=0,\quad\forall y\geq 0$
$\displaystyle u(x,0)=0,\,\lim_{y\to +\infty} u(x,y)=...

**2**

votes

**1**answer

289 views

### $L^p$ compactness for a sequence of functions from compactness of 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(...

**0**

votes

**0**answers

38 views

### An example of a certain continuous and strictly convex function

Let $X$ be a locally convex topological vector space.
I am looking for an example of a function $f: X \times X \to [0,\infty]$ with the following properties:
(1) For all $x,y \in X$, $f(x,y) = 0$ if ...

**9**

votes

**1**answer

159 views

### Literature request: Schatten class difference of semigroups

Let $\mathcal{H}$ be a Hilbert space and $A,B$ two operators on it (not necessarily self-adjoint) such that $A, A+B$ are generators of strongly continuous one parameter semigroups $e^{-tA},e^{-t(A+B)}$...

**1**

vote

**1**answer

122 views

### Regarding subspace generated by the polynomial multiples of outer functions

Let $\mathbb{D}$ and $\mathbb{T}$ denote the open unit disk and unit circle in $\mathbb{C}$ respectively. We write $Hol(\mathbb{D})$ for the space of all holomorphic functions on $\mathbb{D}.$ The ...