Functional
Analysis| Functional
Analysis |
In General
* 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.
@ 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; van Mill 01.
@ For physicists: Boccara 90; Zeidler 95.
Functional Derivative
* 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(
),
with f(0)
= f0, there
exists dA/d at =
0, and it can be expressed as dA/d =
f,
for some distribution ;
Then we call =: A/f,
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].
Specific Spaces and Results > see Hahn-Banach Theorem; Orlicz Space; Sobolev Space.
Functional Equations
@ Differential equations: Hale 77; Tapia NCB(90); Hale & Verduyn
Lunel 93; Antonevich 96 [operator approach]; Czerwik 02 [several variables];
Oberlack
& Waclawczyk mp/06 [Lie
group techniques].
@ Second-order: Golinski & Odzijewicz mp/02 [factorization
method].
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
14 jun 2008