Functional
Analysis |

**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.

**Functional Derivative**

* __Idea__: The Fréchet
derivative of a functional.

$ __Def__: A functional *A*[*f*]
is functionally differentiable at *f*_{0} if
for any 1-parameter family of functions *f*(*λ*),
with *f*(0) = *f*_{0}, there
exists d*A*/d*λ* at *λ* = 0, and it can be expressed as d*A*/d*λ* =
∫ χ δ*f*, for some distribution *χ*;
Then we call *χ* =: δ*A*/δ*f*,
the functional derivative of *A* at *f*_{0}.

* __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
Spaces.

@ __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].

**Functional Equations**

@ __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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send feedback and suggestions to bombelli at olemiss.edu – modified
9 may 2017