Explore: Banach Spaces

Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. A Banach space is

that bear Banach's name include Banach spaces, Banach algebras, Banach measures, the Banach–Tarski paradox, the Hahn–Banach theorem, the Banach–Steinhaus

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

and the normed space is a Banach space. Those spaces for which the canonical evaluation map is surjective are called semi-reflexive spaces. In 1951, R.

vector space can be "uniquely extended" to a Banach space, which makes normed spaces intimately related to Banach spaces. Every Banach space is a normed

defined general Banach spaces and used Helly's technique to prove a norm-preserving version of Hahn–Banach theorem for Banach spaces (where a bounded

spaces (modules over a field) and abelian groups (modules over the ring Z of integers). The construction may also be extended to cover Banach spaces and

Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that

infinite-dimensional normed vector spaces may or may not be complete; those that are complete are Banach spaces. The space C[a, b] of continuous real-valued

require that X , Y {\displaystyle X,Y} are Banach, but the definition can be extended to more general spaces. Any bounded operator T {\displaystyle T}