Functional analysis

Functional Analysis is a branch of mathematics that studies vector spaces endowed with some kind of limit-related structure (e.g., inner product, norm, topology) and the linear functions defined on these spaces and respecting these structures in a suitable sense. It has its historical roots in the study of functional spaces, particularly transformations of functions, such as the Fourier transform, as well as in the study of differential and integral equations. This field of mathematics is related to many other areas of mathematics and science, particularly calculus of variations, complex analysis, and quantum mechanics.

Definition and Scope

Functional analysis focuses on the study of spaces of functions and their mapping properties. The primary objects of interest are the Banach spaces and Hilbert spaces. Banach spaces are vector spaces equipped with a norm that turns them into a complete metric space. Hilbert spaces, a subset of Banach spaces, are endowed with an inner product that allows for a geometric interpretation of vector space concepts.

Key Concepts

Normed Spaces and Banach Spaces

A normed space is a vector space V over the field ℝ or ℂ, together with a norm function that assigns a non-negative scalar to each vector. A normed space is complete (i.e., all Cauchy sequences in the space converge) is called a Banach space.

Inner Product and Hilbert Spaces

An inner product space is a vector space with an additional structure called an inner product. This inner product allows for the definition of angles and lengths in the vector space. A complete inner product space is known as a Hilbert space, which is central to the study of functional analysis due to its geometric properties.

Linear Operators and Bounded Operators

In functional analysis, a linear operator is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. A linear operator is said to be bounded if there is a constant C such that for all vectors v in its domain, the norm of the image of v is less than or equal to C times the norm of v. The study of bounded operators on Hilbert spaces is a key aspect of functional analysis.

Applications

Functional analysis has numerous applications across mathematics and physics. In quantum mechanics, for example, the states of a quantum system are represented by vectors in a Hilbert space, and physical observables are represented by linear operators on these spaces. In partial differential equations, functional analysis techniques are used to prove the existence and uniqueness of solutions under given conditions.

See Also

• Topology
• Measure Theory
• Spectral theory
• Operator theory