Fréchet Space

A Fréchet space is a complete topological vector space whose topology is defined by a countable family of seminorms rather than a single norm. It occupies the middle ground between Banach spaces (normed and complete) and general topological vector spaces (too permissive for useful analysis): Fréchet spaces retain enough structure to support differentiation, power series, and the implicit function theorem, while being flexible enough to include spaces of smooth functions and distributions that cannot be normed.

Named after Maurice René Fréchet, who pioneered abstract metric and topological spaces in the early 20th century, Fréchet spaces appear naturally in analysis. The space of smooth functions on a compact manifold, the space of holomorphic functions on a domain, and the space of test functions in distribution theory are all Fréchet spaces. Unlike Hilbert spaces or Banach spaces, a Fréchet space may not have a norm that induces its topology; instead, convergence is defined by simultaneous convergence in all seminorms.

Fréchet spaces are often treated as a technical footnote between normed spaces and general topological vector spaces. This is backwards. The spaces that actually arise in differential equations and physics — spaces of smooth functions, spaces of distributions, spaces of holomorphic functions — are Fréchet, not Banach. The norm is a convenience; the seminorm family is the reality.