Banach Fixed Point Theorem

Let $f$ be a contraction mapping from a closed Subset $F$ of a Banach Space $E$ into $F$. Then there exists a unique $z\in F$ such that $f(z)=z$.

See also Fixed Point Theorem

References

Debnath, L. and Mikusinski, P. Introduction to Hilbert Spaces with Applications. San Diego, CA: Academic Press, 1990.