Definition:Banach Space

Definition

A Banach space is a complete normed vector space.

That is, a Banach space is a normed vector space where every Cauchy sequence is convergent.

Also defined as

Some sources additionally specify that the normed vector space over which a Banach space is defined also has to be over the real or complex numbers.

This distinction is of limited importance, as it is vanishingly rare for a Banach space to be used in a different context.

Examples

Euclidean Space

A Euclidean space with the usual Euclidean norm is a Banach space.

Square-Integrable Real-Valued Mappings

Let $S$ be the space of all square-integrable real-valued functions.

Then $S$ is a Banach space.

Also see

• Definition:Hilbert Space
• Definition:Functional Analysis

• Results about Banach spaces can be found here.

Source of Name

This entry was named for Stefan Banach.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Banach space
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Banach space
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Banach space
• 2013: Francis Clarke: Functional Analysis, Calculus of Variations and Optimal Control: $5$: Banach spaces
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Banach space
• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.4$: Normed and Banach spaces. Sequences in a normed space; Banach spaces
• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $4.1$: Banach Spaces
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Banach space