# Questions tagged [fa.functional-analysis]

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

5,346 questions

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186 views

### Generalization of Carleman coefficients to multivariable functions - Carleman tensor?

Recently I learned about a matrix called
Carleman matrix. It is a matrix used to represent function iteration with matrix multiplying.
Carleman linearization is a technique used to embed a finite
...

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131 views

### The determinant curvature

Let $(M,g)$ be a riemannian manifold and $R(X,Y)$ the riemannian curvature as a two form with values in the endomorphisms of the tangent bundle. I define:
$$
D_g(X,Y)=det(R)(X,Y)
$$
with $det$ the ...

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140 views

### What functions can one try employing to fit an apparently doubly-periodic real function over $[0,1]$?

I have a cosine-like data curve over $x \in [0,1]$ that I can rather well-fit by
a function of the form $a \cos{2 \pi x} +b$. Although good, the fit is still lacking,
in that the residuals from the ...

**3**

votes

**1**answer

81 views

### Measurability of specific function

Let $I\subset\mathbb{R}$ denote an open and bounded interval of the real line, $H_0^1(I)$ all quadratic integrable Sobolev functions and $C(\bar{I})$ all continuous functions on said interval.
Since ...

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47 views

### Necessary conditions for a function to be represented as symmetric tensor product

Let $f(\cdot,\cdot)$ be any function of 2 arguments. Suppose also that the following equation holds $$f(y,x)+f(x,y)=g(x) h(y) +g(y)h(x) \quad \forall x, y$$ for arbitrary $h(\cdot)$ and $g(\cdot)$. ...

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294 views

### Does point-wise weak convergence give weak convergence in $L^2(I;X)$?

Let $X$ be a separable reflexive Banach space, $F$ be a locally Lipschitz nonlinear operator on $X$ that is weakly continuous on $X$, and $u_n$ are $u_n$ weakly converges to $u$ on $L^2(0,T;X)$. Now, ...

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128 views

### Criteria for Schrödinger operator on real line to have simple spectrum

Consider a Schrödinger operator $H:=-\Delta+V$ on $\mathbb R$, where $V$ is such that $H$ has a purely discrete spectrum
$-\infty<\lambda_1\leq\lambda_2\leq\cdots$ converging to $+\infty$. Do there ...

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80 views

### What is a relation between energy space and $L^p_s-$Sobolev spaces?

We define energy space
$$E= \left\{ f\in \mathcal{S}'(\mathbb R^d): \|\nabla f\|_{L^2} + \|xf\|_{L^2} < \infty \right\}.$$
and
Sobolev spaces
$$L^p_s(\mathbb R^d)=\{f\in \mathcal{S}'(\mathbb R^...

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**1**answer

106 views

### Uniqueness of solution to system of integral equations

Given the following system of integral equations for an integrable function $f(x)$:
For all integers $k \ge 1$ holds
$\int_{0}^{2\pi} [f(x)]^k e^{(ikx)} dx = 0$.
If $f(x)$ is real-valued and non-...

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41 views

### Distributions and interchange of integrals under a PDE

For some $\eta>\epsilon>0$ (small), I have a double (actually many-dimensional) integral of the form
$$ F(x) = F(f;x) = \int_{\Bbb{R}} \int_{\Bbb{R}} f(y) h(x,y,z) dy dz, \qquad x>0$$
where $...

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60 views

### Clifford algebras in the context of locally convex topological vector spaces

Suppose given a locally convex Hausdorff topological vector space $V$ over $\mathbb R$ and a continuous, symmetric, bilinear map $q:V\otimes V\to \mathbb R$, where the tensor product is the completed ...

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**1**answer

66 views

### Approximate Selection for finite-valued Upper Hemicontinuous/Semicontinuous Maps?

I'd like to know if there are any known-results on the existence of continuous approximation theorems for upper hemicontinuous (aka upper semicontinuous) maps $\phi: X\rightarrow Y$ which are finite ...

**2**

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**1**answer

49 views

### Basic sequence vs uniform boundedness of projections

Let $\{e_n,e^*_n\}_{n=1}^\infty$ be a biorthogonal system in $\mathbb{X}\times\mathbb{X}^*$. The system $\{e_n\}_{n=1}^\infty$ is called a basic sequence if it is a Schauder basis of the subspace $E={\...

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**1**answer

82 views

### Orthogonal complement vector space

Let $X$ be a vector space contained in $H^{1}(\mathbb R^d),$ then we can study
$X^{\perp_{L^2}}:=\left\{ \xi \in L^2; \langle \xi, x \rangle_{L^2} =0 \ \forall x \in X \right\}$
and
$X^{\perp_{H^{-...

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59 views

### Common eigenvector of commuting unbounded Operators

Let be T,S two self adjoint linear Operator on a Hilbert Space $\mathcal{H}$ with pure point spectrum.
Then T,S commute if and only if they have a complete set of common eigenvectors.
This is the ...

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**1**answer

157 views

### Is the kernel of a Fredholm operator stable under perturbation?

This is a follow-up of this question.
In a nutshell: Does the kernel of a bounded operator change "nicely" with the operator?
Let $(X,\| \cdot \|)$ be an infinite-dimensional normed vector space.
...

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185 views

### Which subspaces of $\ell_p^n$ are isometric?

This question is similar to the one asked here:
Extending linear isometries from subspaces of $\ell_p^n$
Let $p$ be an even integer. Let $X,Y$ be subspaces of $\ell_p^n$, and let $U : X \to Y$ be a ...

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77 views

### Conditional Expectation for von Neumann algebra

Let $M$ be a von Neumann algebra and $T: M\rightarrow \mathbb{C}$ be a finite normal faithful tracial map, s.t., if $\phi : M \rightarrow A \cap A^{*}$ is a conditional expectation, (A being a weak ...

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58 views

### Reference Request: Egoroff Theorem for nets

Does there exist a generalization of Egoroff theorem for nets instead of sequences of functions?

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55 views

### Dense Egoroff theorem

Suppose that $f_n:X\rightarrow V$ is a sequence of continuous functions from a compact metric space $X$ to a Banach space $V$ and let $\mu$ be a Radon measure on $X$ and $\epsilon>0$ be given.
...

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**1**answer

92 views

### small perturbation of BV function

consider an interesting real analysis question:
define average operator on $[0,1]$:
$A_{\epsilon} f (x) = \frac{1}{2\epsilon}\int_{x-\epsilon}^{x+\epsilon} f(y) dy , f \in BV[0,1] $
( may clarify ...

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**1**answer

185 views

### Can this self-adjoint operator have an infinite-dimensional compression with compact inverse?

The following might be quite straightforward, but I very rarely work in detail with unbounded operators, so I thought it would be worth seeing quickly if I have overlooked an example that is obvious ...

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106 views

### Closure of tensor product /tensor product semigroup

In this reference the following claim is made in Remark 2
Let $A,B$ be closable operators on Banach spaces $X,Y$, then $A \otimes 1$ and $1 \otimes B$ are closable operators on the Banach space $X \...

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167 views

### On convergent sequences in locally convex topological vector spaces

Assume that a sequence $(x_n)_{n\in\omega}$ of points of a locally convex topological vector space converges to zero. Is it always possible to find increasing number sequences $(n_k)_{k\in\omega}$ and ...

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33 views

### Invariance of simple functions

Let $(A(s),D(A(s)))_{s\in\mathbb{R}}$ be a family of unbounded operators on a Banach space $X$ and $g:\mathbb{R}\rightarrow X$ be a simple function, i.e.,
\begin{align*}
g=\sum_{i=1}^n{x_i\textbf{1}_{...

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75 views

### An example of a sequence of finite projections

Let $A$ be a vn-algebra. Suppose that $x$ is an isometry with $\inf_{n\geq1} x^nx^{*n}=0$ (Note that $x^nx^{*n}$ are all projections). Let $e$ be a (non-zero) finite projection and put $q_n$ to be ...

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81 views

### An example of a particular vn-algebra

Let $A$ be a vn-algebra. Let us suppose $e$ is a finite projection in $A$ and $x$ is an isometry (meaning $x^*x=1$) in $A$ such that $e$ does not commute with $x$. Then $\{q_n=x^nex^{*n}\}$ forms a ...

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879 views

### A Question on certain Hilbert space of continuous functions, and a characteristic of convergence in it

Define $T^k(\Omega)$, $\Omega$ an open subset of $\mathbb{R}^m$ (with a smooth boundary), as a space of function equivalance classes, with the norm defined as $$ \|f\|_{T^k(\Omega)}^2 = \|f\|_{L^2(...

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165 views

### Sobolev space under Mellin transform

The Mellin transform is known to be an isomorphism see wikipedia
between $M:L^2(0, \infty) \rightarrow L^2(-\infty, \infty)$
where $$M(f):= \frac{1}{\sqrt{2\pi}}\int_0^{\infty} x^{-\frac{1}{2} + is} ...

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**1**answer

55 views

### The range projection of product of projections

Let $A$ be a von Neumann algebra. Let $p$ be a projection in $A$. Suppose that $e$ is a finite projection. Can we determine all types of vn-algebras in which $p-p\wedge(1-e)$ is a finite projection?...

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55 views

### A question on existence of a Sobolev Hilbert space, where convergence implies uniform convergence [closed]

Is there a Sobolev Hilbert space $H^k(\Omega)$($\Omega$ open subset of $\mathbb{R}^m$, with a smooth boundary), for some $k \in \mathbb{N}$, such that, any sequence in the space $C^0(\bar{\Omega})\cap ...

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46 views

### Unclear inequality of L2 norms (Poisson equation for modeling flow)

I encountered a problem working through a paper about modeling flow with the use of the Poisson equation (source given below). There appears an inequality of L2 norms I don't understand so far. Your ...

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243 views

### Can a positive polynomial on sphere be represented as the sum of squares of spherical harmonics

Let $p\in {\mathbb{R}}[x_1,\ldots, x_d]$ be a homogenous polynomial degree $2n$. We know that if $p$ is positive on $[-\pi,\pi]^d$, $p$ is sum of squares polynomial, i.e. $p$ can be witten as sum of ...

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58 views

### Convex hull of piece-wise linear functions

Let $K>1$ be a positive integer. Consider a function class $$\mathcal{F}_K:=\Big\{\max_{1\leq k\leq K} a_k^\top x + b_k:\ ||a_k||\leq 1, |b_k|\leq 1, \forall 1\leq k\leq K \Big\}$$ on some compact ...

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**1**answer

65 views

### Regarding $\ell_p$ direct sums

I am reading this paper by S.H Karin titled Norm attaining operators and pseudospectrum.
In page 2 he gives the definition of $l_p$ direct sum of a family of Banach spaces as follows:
If $1\leq p< \...

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57 views

### Show that the norm's bound is an exponent

Let $f$ be a continuous function on $S^2$. Consider $g\in C^{\infty}(R)$, such that $g(x)=1$ for $|x|\leq 1$ and for $|x|\geq 2$. Let $h(x)=g(x)-g(2x)$. The notation $proj_k$ denotes the orthogonal ...

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109 views

### Operator power of another operator

I was reading a paper and encountered the following notation:
Let $\mathcal{H}=\ell^2(\mathbb{Z})$ and $\{e_p\}_{p\in \mathbb{Z}}$ be an orthonormal basis of $\mathcal{H}$.Define
$$ue_p=e_{p+1}\quad ...

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295 views

### Hölder continuity for operators

Let $x,y$ be positive real numbers then
$$|\sqrt{x}-\sqrt{y}|=\dfrac{|x-y|}{\sqrt{x}+\sqrt{y}}=\sqrt{|x-y|}\cdot \dfrac{\sqrt{|x-y|}}{\sqrt{x}+\sqrt{y}}\leq 1\cdot |x-y|^{\frac{1}{2}}$$
we obtain $1/...

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491 views

### Structures of the space of neural networks

A neural network can be considered as a function
$$\mathbf{R}^m\to\mathbf{R}^n\quad
\text{by}\quad x\mapsto w_N\sigma(h_{N-1}+w_{N-1}\sigma(\dotso h_2+w_2\sigma(h_1+w_1 x)\dotso)),$$
where the $w_i$ ...

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121 views

### A problem on integrability of derivatives

Let $$f : (0,1) \to \mathbb{R}$$ and $$g(x) = |f(x)|^{r-1} f(x)$$$r \in \mathbb{N}$. It is known that $g\in \mathcal{L}^2(0,1)$ and the $r^{th}$ weak derivative, $ g^{(r)} \in \mathcal{L}^2(0,1)$. I ...

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246 views

### Is that correct $\mathbb R^2\cong\mathbb R$ as measurable spaces? [closed]

Is that correct $R^2\cong R$ as measurable spaces?
If we consider $R$ and $R^2$ with Borel $\sigma$-algebras, is there measurable map from $R$ to $R^2$ with measurable inverse?

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64 views

### Quasinilpotent operator in finite von Neumann algebra

If the trace of all positive powers of a $n \times n$ complex matrix is $0$, then the matrix must be nilpotent. https://math.stackexchange.com/questions/159167/traces-of-all-positive-powers-of-a-...

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**1**answer

192 views

### Schwartz space on $\bigcup_{n=1}^CR^n$

I have an application where I need to work with the following idea.
Let the space $\bigcup_{n=1}^C \mathbb{R}^n$ be associated with the metric $d$ such that for $x=(x_1,\cdots,x_n)$ and $y=(y_1,\cdots,...

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326 views

### One question about the $\eta$ invariant

This question is from the paper, The Analysis of Elliptic Families
II. Dirac Operators, Eta Invariants, and the Holonomy Theorem, Commun. Math. Phys. 107, 103-163 (1986) --- Proposition 2.8.
Suppose ...

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142 views

### Improving equi-integrability for a family $\mathcal F$ in $L^1(\Omega)$

Let $\mathcal F$ be a weakly compact subset of $L^1(\Omega)$. Dunford–Pettis theorem says that $\mathcal F$ is uniformly integrable. Also, by de la Vallée-Poussin theorem we can find an increasing ...

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57 views

### A conjecture characterizing almost uniform convergence of finitely additive conditional probabilities

This question is a continuation of a question I asked a couple weeks ago.
Let $(\Omega, \mathcal{C})$ be the Cantor space of binary sequences equipped with the usual product topology, and let $(\...

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**1**answer

73 views

### Solve nonlinear equation

Suppose that $f:E\to F$(between Banach spaces), is of the form
$$f(x)=f(0)+D(x)+N(x).$$
Here $D$ is a linear term, whose kernel is of finite dimension, and admits a right inverse $G$, i.e. $D(G)(\...

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55 views

### Constant in trace theorem for balls

Consider the standard open ball $B_r:=\left\{x ; \left\lvert x \right\rvert \le R \right\}.$
The trace theorem tells us any function in $W^{k,p}(B_r)$ can be restricted to a function $W^{k-1,p}(\...

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**1**answer

98 views

### A relative Kuiper theorem

Let $(H_0, \langle \,,\,\rangle_0)$ be a real separable Hilbert space,
and let $(H_1, \langle \,,\,\rangle_1)$ be a Hilbert space such that $H_1 \subset H_0$ is dense and such
that the inclusion $(...

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58 views

### Nonlinear maps in Riesz Thorin theorem

The Riesz Thorin theorem allows us to interpolate between $L^p$ spaces and the usual assumption is that the map $T$ is linear.
What I was wondering about is whether this is because otherwise you do ...