A Banach space is a complete normed vector space. It is a mathematical concept that combines the properties of both a vector space and a metric space. In a Banach space, there is a norm defined on the vectors, which gives them a magnitude or length. This norm satisfies certain properties such as non-negativity, triangle inequality, and homogeneity. Completeness is an important property of a Banach space, which means that every [[Cauchy sequence]] in the space converges to a limit within the same space. This property ensures that the space is "filled in" and does not have any "holes" or missing limit points. The concept of Banach spaces was introduced by the mathematician [[Stefan Banach]] in the early 20th century and has since become an important area of study in functional analysis and related fields. Many mathematical objects and functions can be represented as elements of Banach spaces, making them useful tools in various branches of mathematics and physics.