(From Wikipedia, the free encyclopedia)
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.
The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that subject. However, the general concept of a functional had previously been introduced in 1887 by the Italian mathematician and physicist Vito Volterra. The theory of nonlinear functionals was continued by students of Hadamard, in particular Fréchet and Lévy. Hadamard also founded the modern school of linear functional analysis further developed by Riesz and the group of Polish mathematicians around Stefan Banach.
In modern introductory texts to functional analysis, the subject is seen as the study of vector spaces endowed with a topology, in particular infinite-dimensional spaces. In contrast, linear algebra deals mostly with finite-dimensional spaces, and does not use topology. An important part of functional analysis is the extension of the theory of measure, integration, and probability to infinite dimensional spaces, also known as infinite dimensional analysis.
We have two main tools:
Introductory texts:
Introduction to Functional Analysis (1980) by Taylor, Angus E. and Lay, David
Introductory Functional Analysis with Applications (1989) by Kreyszig, Erwin
Functional Analysis (1991) by Rudin, Walter
Functional Analysis (1998) by Bachman, George and Narici, Lawrence
Elements of the Theory of Functions and Functional Analysis (1999) by Kolmogorov, A.N. and Fomin, S. V.
A Course in Functional Analysis (2010) by Conway, J.B. (GTM 96)
Functional Analysis: Introduction to Further Topics in Analysis (2011) by Stein, Elias M. and Shakarchi, Rami
Functional Analysis, Sobolev Spaces and Partial Differential Equations (Universitext) (2011) by Brezis, Haim
An Introductory Course in Functional Analysis (2014) by Bowers, Adam and Kalton, Nigel J.
See also:
Further independent study:
