Completion of a normed space

Summary

The completion of a normed space $X$ is a Banach space that contains a copy of $X$ as a dense subspace, with the same norm. It exists and is unique up to isometric isomorphism. It lets one prove results for Banach spaces and apply them to incomplete spaces. Many important spaces, such as $L^p$ spaces and Sobolev spaces, can be built as completions of spaces of nice functions.