Functional analysis
Introduction
Functional analysis is a branch of mathematical analysis that uses the concepts of a vector space to study mathematical functions. The field has its roots in the study of transformations, such as the Fourier transform, and in the study of differential and integral equations. This approach makes it possible to apply the methods of linear algebra to problems in analysis.
History
The development of functional analysis is closely connected with the study of differential equations and the development of quantum mechanics. In the early 20th century, mathematicians such as Hilbert and Banach began to develop the theory of Banach spaces, which are complete normed vector spaces. These spaces provide the setting for the study of more general and abstract differential equations.
Basic Concepts
Functional analysis revolves around the study of spaces of functions. It introduces the notion of various types of measures, spaces, and operators. It also introduces the notion of a topology, making it a direct generalization of linear algebra.
Vector Spaces
A vector space is a fundamental concept in functional analysis. It is a set of elements, called vectors, which can be added together and multiplied by scalars.
Normed Spaces
A normed space is a vector space with a norm, a function that assigns a strictly positive length or size to each vector in the vector space.
Banach Spaces
A Banach space is a complete normed vector space. This means that every Cauchy sequence of vectors in the space converges to a limit within the space.
Hilbert Spaces
A Hilbert space is a complete inner product space - an expansion of the notion of Euclidean space to include infinite dimensions. It is named after David Hilbert, who studied their properties in the early 20th century.
Applications
Functional analysis has broad applications in mathematics and engineering. It is used in the study of differential and integral equations, quantum mechanics, and in many areas of theoretical physics.