# 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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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 $W^{k-1,p}(\... 1answer 98 views ### A relative Kuiper theorem Let$(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$is dense and such that the inclusion$(...
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 ...