The Banach fixed point theorem (also known as the contraction mapping theorem or contraction mapping principle) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self maps of metric spaces, and provides a constructive method to find those fixed points. The theorem is named after Stefan Banach (1892–1945), and was first stated by him in 1922.When using the theorem in practice, the most difficult part is typically to define X properly so that T actually maps elements from X to X, i.e. that T(x) is always an element of X.

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