Search results
Results from the WOW.Com Content Network
The basic and historically first class of spaces studied in functional analysis are complete normed vector spaces over the real or complex numbers. Such spaces are called Banach spaces . An important example is a Hilbert space , where the norm arises from an inner product.
In mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space and one-parameter families
This is a list of functional analysis topics. See also: Glossary of functional analysis. Hilbert space. Bra–ket notation; Definite bilinear form; Direct integral;
The study, which depends heavily on the topology of function spaces, is a branch of functional analysis. If a collection of operators forms an algebra over a field, then it is an operator algebra. The description of operator algebras is part of operator theory.
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 operators acting upon these spaces and respecting these structures in a suitable sense.
In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm. The theorem was first published in 1927 by Stefan Banach and Hugo Steinhaus, but it was also proven independently by Hans Hahn.
Functional principal component analysis (FPCA) is a statistical method for investigating the dominant modes of variation of functional data.Using this method, a random function is represented in the eigenbasis, which is an orthonormal basis of the Hilbert space L 2 that consists of the eigenfunctions of the autocovariance operator.
The spectrum of a linear operator that operates on a Banach space is a fundamental concept of functional analysis. The spectrum consists of all scalars λ {\displaystyle \lambda } such that the operator T − λ {\displaystyle T-\lambda } does not have a bounded inverse on X {\displaystyle X} .