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];
Sasane 17 [friendly].

@ __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 JMP(18)-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 13 jun 2018