# Questions tagged [fa.functional-analysis]

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

6,405
questions

**-1**

votes

**0**answers

63 views

### The space of harmonic functions on an open set is infinite dimensional? [closed]

I want to prove that he space of harmonic functions on an open set $\Omega \subset \mathbb{R}^n $ , with $n \geq 2$, is uncountablely infinite-dimensional.
I guess that I have to find a linearly ...

**2**

votes

**1**answer

59 views

### Uniform integrability contradicts convergence to $L^2$ subspace

The following question was asked at https://mathoverflow.net/questions/361367/uniform-integrability-contradicts-convergence-to-l2-subspace :
Let $V$ be a finite-dimensional subspace of $L^2(\...

**2**

votes

**1**answer

63 views

### Approximation of vectors using self-adjoint operators

Let $T$ be an unbounded self-adjoint operator.
Does there exist, for any $\varphi$ normalized in the Hilbert space, a constant $k(\varphi)>0$ and a sequence of normalized $(\varphi_n)$ such that $$...

**1**

vote

**0**answers

54 views

### G-abelian systems

Let $(\mathfrak{A},\alpha,\phi)$ be a $C^*$-dynamical system made of a unital $C^*$-algebra, a $*$-automorphism and an extremal invariant (i.e. ergodic) state.
Consider the covariant GNS ...

**4**

votes

**0**answers

56 views

### Characterization of “PSD-Squared” Matrices

$\DeclareMathOperator\DNN{DNN}\DeclareMathOperator\CP{CP}$This question can be thought of as an offshoot of this MO question from a few months ago. Let $M_n(\mathbb{C})$ denote the set of $n \times n$ ...

**2**

votes

**0**answers

52 views

### General construction of enveloping C*-algebra, left/right-regular representation, etc

In a number of contexts (e.g. groups, crossed products, groupoids, Fell bundles) there are similar constructions of enveloping C*-algebras and left/right-regular representations that incorporate ...

**2**

votes

**0**answers

41 views

### Holomorphic semigroups vs analytic semigroups

Is there any difference between the two notions in the theory of semigroups?
In the literature, we find some monographs use the farmer while others use the latter. I expect that they are always the ...

**6**

votes

**3**answers

707 views

### Are nuclear operators closed under extensions?

Given $X_i, Y_i$ Banach spaces, $f_j, g_j, T_i$ bounded linear operators for $i=1,2,3$ and $j=1,2$. We have the following diagram
$\require{AMScd}$
\begin{CD}
0 @>>> X_1 @>f_1>> X_2 ...

**4**

votes

**1**answer

121 views

+50

### Riesz transform of fractional operators

I am interested in Riesz transforms linked to the fractional Laplacian and other fractional operators. I have been hunting down in the literature to find related results but I have not been able to ...

**2**

votes

**1**answer

47 views

### Classes of Banach algebras (that aren't operator algebras) whose bidual comes from a “universal representation”

Are there any classes of (Arens regular) Banach algebras that are not operator algebras whose bidual comes from a “universal representation”, as in the case of C*-algebras?

**3**

votes

**1**answer

74 views

### Generalized tensor-train decomposition

If $U\in\bigotimes_{k=1}^p\mathbb R^{n_k}$ and $U^{(k)}\in\mathbb R^{r_{k-1}}\otimes\mathbb R^{n_k}\otimes\mathbb R^{r_k}$ with$^1$ $$U_{i_1,\:\ldots\:,i_p}=\sum_{j_0=1}^{r_0}\cdots\sum_{j_p=1}^{r_p}U^...

**1**

vote

**1**answer

58 views

### Eigenspace of Gaussian Markov operator

Consider the (one-dimensional) Gaussian distribution $Q := N(\nu,\tau^2)$ and the (Gaussian) Markov operator
\begin{equation*}
\begin{array}{rccc}
R : & L_1(\mathbb{R},\mathcal{B}(\mathbb{R}),Q) &...

**0**

votes

**0**answers

38 views

### Tensor contraction (vector-valued trace) on $\bigotimes_{i=1}^k\mathcal L(E_{i-1},E_i)$

If $E_i$ is a $\mathbb R$-vector space, then the vector-valued trace $\operatorname{tr}_{E_1}:(E_2\otimes E_1^\ast)\otimes(E_1\otimes E_0)\to E_1\otimes E_0$ (or tensor contraction) is the ...

**-4**

votes

**0**answers

51 views

### partial differential inequality [duplicate]

I want to prove that if $f\leq l$ and $\lim_{a\to \infty}\frac{f(a)}{l(a)}=1$ (we can also suppose that $\lim_{a\to \infty}f(a)=\lim_{a\to \infty}l(a)=1$) where $f$ is a positive continuous bounded ...

**5**

votes

**2**answers

175 views

### Surjection in compact-open topology [closed]

Let $Z$, $X$ and $Y$ be topological spaces and let $f:X\to Y$ be a continuous surjection then is the induced map $g \to f\circ g$ from $C(Z,X)$ to $C(Z,Y)$ is continuous. But is it still a surjection?...

**4**

votes

**4**answers

171 views

### On Köthe sequence spaces

I asked this a week ago at math.stackexchange, but without success.
As far as I understand, there are several meanings of the notion of the Köthe sequence space, in particular, Hans Jarchow in his "...

**4**

votes

**0**answers

68 views

### Smoothing continuous functions in metric space

Let $(X,\rho)$ be a metric space.
For any $f:X\to\mathbb{R}$, define the local Lipschitz constant of $f$ at $x$ by
$$ \Lambda_f(x) := \sup_{x'\in X\setminus\{x\}} \frac{|f(x)-f(x')|}{\rho(x,x,')}
.
$$...

**2**

votes

**1**answer

71 views

### Extension of outer unit normal vector to interior

Suppose we have a bounded smooth domain $\Omega$ in $\mathbb{R}^n$, so there exists an outer unit normal vector field $\eta$ everywhere on the boundary. Can we extend it to the interior satisfying ...

**1**

vote

**0**answers

38 views

### Operational quantities characterizing upper semi-Fredholm operators

An operator $T:X\rightarrow Y$ is said to be upper semi-Fredholm if its range is closed and its kernel is finite-dimensional. M. Schechter (1972) introduced a quantity $$\nu(T):=\sup_{\operatorname{...

**1**

vote

**0**answers

14 views

### Show that a tensor-train is contained in a recursive sequence of subspaces

Let
$p\in\mathbb N$;
$n_k\in\mathbb N$ and $\left(e^{(k)}_1,\ldots,e^{(k)}_{n_k}\right)$ denote the standard basis of $\mathbb R^{n_k}$ for $k\in\{1,\ldots,p\}$;
$u\in\bigotimes_{k=1}^p\mathbb R^{n_k}...

**4**

votes

**0**answers

68 views

### Equality in spectral inclusion theorem

I asked this question on Math SE but didn't receive any response.
Let $(T_t)$ be a $C_0$-semigroup on a Banach space $X$ with generator $A.$ If $\lambda_0\in \mathbb{C}$ is such that $e^{\lambda_0 t}$...

**0**

votes

**0**answers

30 views

### How are the vector-valued trace and the unique linearization of $\mathfrak L(X,Y)\:\hat\otimes_π\:X→Y$ of $\mathfrak L(X,Y)×X→Y,\;(L,x)↦Lx$ related?

Let $X$ be a $\mathbb R$-Banach space and $X'\:\hat\otimes_\pi\:X$ denote the completion of the tensor product of $X'$ and $X$ with respect to the projective norm. The trace functional $\operatorname{...

**0**

votes

**1**answer

44 views

### Cores in the tensor-train decomposition

Let $d_i\in\mathbb N$, $I_i:=\{1,\ldots,d_i\}$ and $u\in\mathbb R^{d_1}\otimes\mathbb R^{d_2}\otimes\mathbb R^{d_3}$. It's somehow clear to me that we may regard $u$ as a three-dimensional array (see ...

**0**

votes

**0**answers

32 views

### A functional that occurs in Vlasov-Poisson equation

Let me share a functional that pops up in the analysis of the Valsov-Poisson equation (see the motivation below). At given time, the macroscopic mass density is $x\mapsto\rho(x)\ge0$. Assuming finite ...

**-1**

votes

**0**answers

49 views

### Solving Problems of product Series [closed]

Is there any general method to solve various types of Product Series Problems including the pi product forms?

**1**

vote

**1**answer

64 views

### Convergence of the regularized gradient of a Lipschitz function

Let $\varphi:\mathbb R^d\to\mathbb R_+$ be given as
$$
\varphi(x) := \begin{cases}
c\exp\big(1/(|x|^2-1)\big) & \mbox{if } |x|\le 1 \\
0 & \mbox{otherwise},
\end{cases}
$$
...

**3**

votes

**1**answer

93 views

### Is the closure of the ball of $1$-Lipschitz functions still equi-Lipschitz?

$\DeclareMathOperator\Lip{Lip}$Let $\Lip_0(\mathbb R^d)$ be the space of Lipschitz functions $f:\mathbb R^d\to\mathbb R$ vanishing at zero, i.e., $f(0)=0$, and equipped with the norm $\|f\|:=\|\nabla ...

**-2**

votes

**0**answers

23 views

### How to calculate score based on quantity and quality? [closed]

I want to give each user a correctness score based on number of times he/she was correct and total number of guesses that she made. One user might have 1 guess that was correct, I don't want to give ...

**11**

votes

**1**answer

293 views

### Riesz–Markov–Kakutani representation theorem for compact non-Hausdorff spaces

Let $X$ be a compact Hausdorff topological space, and $\mathcal C^0 (X) = \{f:X\to\mathbb{R}; \ f \text{ is continuous }\}$. It is well known that for any bounded linear functional $\phi: \mathcal C^...

**4**

votes

**1**answer

111 views

### Decay of Fourier coefficients of real analytic functions

I would like to have any suggestion/reference to the following question. I have a differential operator $\mathcal{L}$ with discrete spectrum defined on a a suitable Sobolev space on a domain $\Omega$, ...

**0**

votes

**1**answer

65 views

### A time dependent variational problem coming from a second order linear PDE

Fix $u_0\in H^1(\Omega)$ and $f=f(x,y,t)\in L^2(\Omega\times [0,T])$ where $\Omega$ is a sufficiently smooth bounded domain in $\mathbb{R}^2$.
Consider the problem of finding $u:\Omega\times[0,T]\to\...

**4**

votes

**0**answers

102 views

### Continuous disintegration

Given a suitable Borel measure $\mu$ on a suitable topological space $X$ and a Borel function $\pi:X \to Y$, where $Y$ is another suitable topological space, the disintegration theorem gives a Borel ...

**-1**

votes

**0**answers

40 views

### Relationship between negative operator eigenvalues

Let $L>0$, $c \in (-1,1)$ and $\varphi \in H_{per}^{2}([0,L])$ be fixed. Define $w:= 1-c^2>0$.
Consider the matrix operator $\mathcal{L}_{R}: H_{per}^{2}([0,L]) \times L_{per}^{2}([0,L]) \...

**1**

vote

**1**answer

43 views

### Proof of universality of Toeplitz algebra

It is well-known that the Toeplitz algebra $\mathcal{T}$ (that I view as concrete subalgebra of $\mathbb{B}(\ell^2(\mathbb{N})$) is the universal algebra generated by an isometry, that is, for any $C^*...

**2**

votes

**0**answers

26 views

### Extension Sobolev functions across of lower dimensional subset

This question may be well-known to experts, but I am trying to get myself a rigorous proof. Consider open set $\Omega=B^n_1(0)\setminus B_1^k(0)$ in $\mathbb{R}^n$. If function $u$ is in $H^1(\Omega)$,...

**1**

vote

**1**answer

92 views

### A question on multiplicity of complex polynomial [closed]

This is not a research level question. But due to some reason I can't ask this question on Math Stack Exchange. So, I am asking this question here.
By definition we know that we can measure the ...

**2**

votes

**0**answers

56 views

### On norms of Boolean functions

Let $f: \mathbb{F}_2^n \rightarrow \{-1,1\}$ be a Boolean function, represented by a $N=2^n$ dimensional vector, $f \in \{-1,+1\}^N$.
Define the Fourier transform of $f$ to be $\hat{f}$, where $$\hat{...

**1**

vote

**1**answer

41 views

### Uniform boundedness of resolvents on the imaginary axis

Let $A \colon \mathcal{D}(A) \subset \mathbb{H} \to \mathbb{H}$ be a closed linear operator in a Hilbert space $\mathbb{H}$, which generates a $C_{0}$-semigroup. Suppose that in a $\varepsilon$-...

**0**

votes

**0**answers

31 views

### How to find a sequence which converge to the fixed random variable

Suppose $\mathscr{A}$ is a convex set on $L^\infty(P)$, $\mathscr{B}$ is the closure of $\mathscr{A}$ under the topology $σ(L^\infty(P),L^1(P))$. For every $X\in\mathscr{B}$, can we find a sequence $\{...

**2**

votes

**0**answers

81 views

### Example of a non-reflexive Banach space and two sequences

Let $(E,\mathcal {A}, \mu ) $ be a finite measure space and $X$ be a Banach space. The set of all Bochner-integrable functions from $E$ into $X$ is denoted by $\mathcal{L}_X^1$.
If $X$ is reflexive, ...

**0**

votes

**1**answer

90 views

### $ \overline{(A-A)}\cap\overline{B}(0,r)\text{ is weakly compact, }\forall r>0 $?

Let $X$ be a separable Banach space and $A$ is a subset of $X$ such that
$$
A\cap\overline{B}(0,r) \text{ is weakly compact, } \forall r>0.
$$
Can we say that :
$$
\overline{(A-A)}\cap\overline{...

**2**

votes

**1**answer

71 views

### Spectral representation of closed operators with finite spectral bound

Assume $A$ is a closed linear operator on a Banach space $X$ and is densely defined. Assume the spectral bound $s(A) = \sup\{Re\lambda: \lambda\in \sigma(A)\}$ is finite. For example, if $A$ is the ...

**6**

votes

**0**answers

111 views

### Quotients of subspaces of $C(\alpha)$

A well known problem, attributed to H. P. Rosenthal, asks whether or not every quotient of $C(\alpha)$, $\alpha$ countable ordinal, is $c_0$-saturated. As it is known, $C(\alpha)$ are $c_0$-saturated ...

**2**

votes

**1**answer

66 views

### On $s$-harmonic functions

Is this statement true?
A bounded positive function $v\in C^{2}(\mathbb R^N)$ decaying to zero which is $s$-harmonic function ($s\in (0, 1)$) outside a ball behaves like $|x|^{2s-N}$ near infinity. ...

**4**

votes

**1**answer

90 views

### Existence of an injective unbounded below operator

Given an infinite dimensional Banach space, can we always find an infinite dimensional Banach space $Y$ and an injective bounded operator $T:X\to Y$ such that $T$ is not bounded below?
If $X^{*}$ is ...

**0**

votes

**1**answer

275 views

### What is the relevant literature –if any– on real-valued functions on sets and their Boolean combinations? [closed]

As part of a project (https://arxiv.org/abs/2004.06745),
I've constructed the following table,
$\left(
\begin{array}{ccc}
\hline
Constraint Imposed & Probability & Quasirandom Estimate \\
...

**7**

votes

**1**answer

335 views

### Basis vs Schauder basis in normed spaces

Following the conventions from Heil: "A Basis Theory Primer" and Albiac, Kalton: "Topics in Banach Space Theory", we might define a basis of an (infinite-dimensional) normed space $V$ as a sequence $(...

**1**

vote

**0**answers

66 views

### What is the closure of this set in $H^1(\mathbb{R}^2)$?

I'm not sure that if this is a difficult question or not. I asked it on MSE and it hasn't been answered and so I thought I might ask it here:
What is (or how can we describe) the closure in $H^1(\...

**5**

votes

**1**answer

230 views

### Existence of injective compact operators

We know that if $X$ is a separable Banach space, then for every infinite dimensional Banach space $Y$, there exists an injective compact operator from $X$ to $Y$.
My query is for every Banach ...

**0**

votes

**1**answer

153 views

### $\frac{\partial}{\partial x}\int_{\mathbb{R}}\frac{1}{\sqrt{2 \pi \varepsilon}}e^{-\frac{(x-y)^2}{2\varepsilon}}l(y)dy\leq C\frac{1}{x}$

Let $l$ be a continuous bounded function ($l$ is not differentiable). I want to prove for $x$ large enough that
$$\frac{\partial}{\partial x}\int_{\mathbb{R}}\frac{1}{\sqrt{2 \pi \varepsilon}}e^{-\...