# Questions tagged [fa.functional-analysis]

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

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I am studying Pick Interpolation and Hilbert Function Spaces by Agler and McCarthy. A kernel $k$ on a set $X$ is said to have $M_{s,t}$ Pick property whenever $x_1,x_2, \ldots , x_n \in X$ and $W_1, ... • 83 9 votes 1 answer 577 views ### Prove J.L. Lions’s Lemma without using Fourier transform When I read the book Linear and Nonlinear Functional Analysis with Applications, I came across J.L. Lions's Lemma (the book doesn't give a proof), which states Let$\Omega \subset \mathbb R^n$be a ... • 521 1 vote 0 answers 65 views ### On calculating the second quantization operator$\Gamma(A)$of the Ornstein-Uhlenbeck operator$A$Let$A$be a self-adjoint operator on a Hilbert space , and let$d\Gamma(A)$be the generator of the second quantization of$A$. Consider the following theorem from Segal's "Non-Linear Quantum ... • 90 3 votes 0 answers 71 views ### Absolute continuity of$t \to \lVert u(t) \rVert^2_{H}$and Gelfand triple : are they equivalent? Let$V$be a separable Banach space and$H$be a separable Hilbert space such that $$V \subset H \subset V'$$ and the inclusion maps are continuous with dense images. Here$...
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In Hilbert space $u$, Let $T_1$,$T_2$ is selfadjoint operator, if exit $c>0$ such that $cI\le T_1\le T_2$, prove $T_1$,$T_2$ have a bounded inverse operator and $c^{-1}I\ge T_1^{-1}\ge T_2^{-1}$. I ...
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### Sufficient condition such that the set of zeros of an analytic function $f:\mathbb{R}^n \to \mathbb{R}$ contains only isolated points

Consider a real- analytic function $f: \mathbb{R}^n \to \mathbb{R}$. We know that zeros of $f$, roughly speaking, live in the low dimensional manifolds. My question: Does a 'reasonable' sufficient ...
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### Can these identities for the Euler-Mascheroni constant be proven?

I stumbled upon these 4 limit/integral identities involving Euler's constant aka gamma (~0.5772). They appear to be valid based on inspection but I have no idea how to prove them. In addition, I have ...
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### Cylindrical Wiener processes or SPDE that can make use of Banach valued rough paths?

Rough paths theory has an often advertised perk that it mostly works for general Banach spaces. I am trying to think of some nice examples that actually use this feature, and am coming up stuck. The ...
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### Trace class operators

There is a notion of trace class operator in a Hilbert space. Is there a notion of trace class operator in arbitrary Banach space? locally convex space? A reference will be helpful.
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### Is the Taylor map continuous?

(Skip to the bolded theorem below for my question, if you'd like) Some context on asymptotic expansions and the Taylor map In the setting of irregular singularities of meromorphic connections on the ...
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### Weak$^\ast$ closure of a countably complete sublattice of the unit ball of $L^\infty(\Omega, \mu)$

This is a reframing of my previous question from a Banach lattice perspective: Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity? The previous question ...
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