A Banach space is a type of mathematical structure used in functional analysis, which is a branch of mathematics. It is defined as a complete normed vector space, meaning that it is a vector space equipped with a norm that allows for the measurement of vector lengths and distances. Completeness ensures that every Cauchy sequence in the space converges to a limit within the same space. Banach spaces are important in various areas of mathematics and applied sciences, as they provide a framework for studying linear operators and functional equations. Examples of Banach spaces include L^p spaces, which are used in analysis and probability theory, and C[0,1], the space of continuous functions on the interval [0,1].