Fréchet space

Summary

A Fréchet space is a locally convex space that is also an F-space: its topology comes from a complete translation-invariant metric. Equivalently, it is a complete Hausdorff space whose topology is generated by a countable family of seminorms. Fréchet spaces generalize Banach spaces and include the main spaces of smooth and holomorphic functions. The Hahn–Banach theorem, the open mapping theorem and the uniform boundedness principle all hold for them.