# Banach space

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In mathematics, Banach spaces (pronounced [ˈbanax]) are one of the central objects of study in functional analysis. Many of the infinite-dimensional function spaces studied in analysis are Banach spaces, including spaces of continuous functions (continuous functions on a compact Hausdorff space), spaces of Lebesgue integrable functions known as Lp spaces, and spaces of holomorphic functions known as Hardy spaces. They are the most commonly used topological vector spaces, and their topology comes from a norm.

They are named after the Polish mathematician Stefan Banach, who introduced them in 1920–1922 along with Hans Hahn and Eduard Helly.[1]

## Contents

### Definition

Banach spaces are defined as complete normed vector spaces. This means that a Banach space is a vector space V over the real or complex numbers with a norm ||·|| such that every Cauchy sequence (with respect to the metric d(x, y) = ||xy||) in V has a limit in V.

### Examples

Throughout, let K stand for one of the fields R or C.

The familiar Euclidean spaces Kn, where the Euclidean norm of x = (x1, …, xn) is given by ||x|| = (∑i=1…n |xi|2)1/2, are Banach spaces.