# Banach Space

Last updated: April 6, 2017

## What Does Banach Space Mean?

A Banach space is a complete normed vector space in mathematical analysis. That is, the distance between vectors converges closer to each other as the sequence goes on. The term is named after the Polish mathematician Stefan Banach (1892–1945), who is credited as one of the founders of functional analysis.

In computer science, the mathematician Shahar Mendelson has used the Banach space in machine learning to bound the errors of machine learning algorithms.

## Techopedia Explains Banach Space

In functional analysis, a Banach space is a normed vector space that allows vector length to be computed. When the vector space is normed, that means that each vector other than the zero vector has a length that is greater than zero. The length and distance between two vectors can thus be computed. The vector space is complete, meaning a Cauchy sequence of vectors in a Banach space will converge toward a limit. As the sequence goes on, the distances between vectors get arbitrarily closer together.

Banach spaces are widely used in functional analysis, with other spaces in analysis being Banach spaces. In computer science, Banach spaces also have been applied to machine learning algorithms to measure the generalization error, or how accurate a machine learning algorithm is. The mathematician Shahar Mendelson in particular has used Banach Spaces to improve the reliability of machine learning algorithms. 