In General > s.a. types
of topological spaces [topologies
on function spaces].
* Idea: The branch of analysis that studies properties of mappings of classes of functions from one topological vector space into another; Some think it should be called topological algebra, but that expression seems to have a more general meaning (> see algebra).
@ General references: Riesz & Nagy 55; Goffmann & Perdrick 65; Maddox 70; Brown & Page 71; Reed & Simon 72, v1; Larsen 73; Rudin 73; Balakrishnan 76; Berger 77; Brown 77; Heuser 82; Conway 90; Zimmer 90; Yosida 95; Pugachev & Sinitsyn 99; van Mill 01; Hansen 06 [and Hilbert space]; Swartz 09 [III]; Weaver 13 [III, separable case]; Hansen 16 [and Hilbert space, 2nd ed].
@ Undergraduate introductions: Pons 14 [II].
@ Special approaches: Ng 10 [non-standard methods].
@ For physicists: Boccara 90; Zeidler 95; Banks 12 [applications in science and engineering]; Brouder et al a1705 [framework and properties]; > s.a. quantum information theory.
* Idea: The Fréchet derivative of a functional.
$ Def: A functional A[f] is functionally differentiable at f0 if for any 1-parameter family of functions f(λ), with f(0) = f0, there exists dA/dλ at λ = 0, and it can be expressed as dA/dλ = ∫ χ δf, for some distribution χ; Then we call χ =: δA/δf, the functional derivative of A at f0.
* Remark: If A[f] is an integral over some fixed domain of integration of an expression involving f(x), then the functional derivative with respecto to f(x) is just the regular derivative of the integrand with respect to f(x).
@ References: Dickey LMP(08) [when the boundary of the domain is not fixed].
@ Generalizations: Tarasov IJAM(14)-a1502 [fractional].
Specific Spaces and Results > s.a. Hahn-Banach
Theorem; hilbert space; Hp Spaces; Orlicz Spaces; Sobolev
@ References: Kundu T&A(10) [metrizability and completeness of the support-open topology on C(X)]; Diening et al 11 [Lebesgue and Sobolev spaces with variable exponents].
@ Differential equations: Hale 77; Tapia NCB(90); Hale & Verduyn Lunel 93; Antonevich 96 [operator approach]; Czerwik 02 [several variables]; Oberlack & Wacławczyk mp/06 [Lie group techniques]; Curtright et al JPA(11)-a1105 [approximate solutions]; Venturi a1604 [numerical methods].
@ Second-order equations: Goliński & Odzijewicz JCAM(05)mp/02 [factorization method].
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