Definition:Banach Space
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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
- 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