Banach space

mathematics

Written by

The Information Architects of Encyclopaedia Britannica

Last updated

June 24, 2026

Quick Summary

A Banach space is a fundamental concept in functional analysis, a branch of mathematics that emerged in the 20th century. Named after Polish mathematician Stefan Banach, it is defined as a complete normed vector space.12

In simpler terms, a vector space is a collection of objects (like numbers or functions) that can be added together and multiplied by scalars. A norm assigns a "length" or "size" to each object in the space. A Banach space is a vector space where this length is well-defined, and crucially, it is "complete." Completeness means that any sequence of objects that appears to be converging actually does converge to an object within the space. This property is essential for many analytical operations, such as differentiation and integration, to work reliably. Banach spaces provide a powerful framework for studying various mathematical problems, including partial differential equations.34

Summary created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

5 Britannica Sources

Ask Britannica AI about this topic

Britannica doesn't have a full article on this topic, so we've curated a list of our relevant coverage.

analysis
Functional analysis
…a Hilbert space and a Banach space, named after the German mathematician David Hilbert and the Polish mathematician Stefan Banach, respectively. Together they laid the foundations for what is now called functional analysis.
Read More
…a Hilbert space and a Banach space, named after the German mathematician David Hilbert and the Polish mathematician Stefan Banach, respectively. Together they laid the foundations for what is now called functional analysis.
Read More

Stefan Banach
…which are now known as Banach spaces. He also proved several fundamental theorems in the field, and his applications of theory inspired much of the work in functional analysis for the next few decades.
Read More
…which are now known as Banach spaces. He also proved several fundamental theorems in the field, and his applications of theory inspired much of the work in functional analysis for the next few decades.
Read More

Timothy Gowers
…of several outstanding problems of Banach spaces. His dichotomy theorem asserts that either every subspace of a given Banach space has many symmetries or the subspaces have only trivial symmetries. He also did profound work on combinatorial number theory and gave an improved proof of number theorist Endre Szeméredi’s theorem…
Read More
…of several outstanding problems of Banach spaces. His dichotomy theorem asserts that either every subspace of a given Banach space has many symmetries or the subspaces have only trivial symmetries. He also did profound work on combinatorial number theory and gave an improved proof of number theorist Endre Szeméredi’s theorem…
Read More