Fréchet manifold

In mathematics, in particular in nonlinear analysis, a Fréchet manifold is a topological space modeled on a Fréchet space in much the same way as a manifold is modeled on a Euclidean space.

More precisely, a Fréchet manifold consists of a Hausdorff space X with an atlas of coordinate charts over Fréchet spaces whose transitions are smooth mappings. Thus X has an open cover {U_α}_{α ε I}, and a collection of homeomorphisms φ_α : U_α → F_α onto their images, where F_α are Fréchet spaces, such that

${\displaystyle \phi _{\alpha \beta }:=\phi _{\alpha }\circ \phi _{\beta }^{-1}|_{\phi _{\beta }(U_{\beta }\cap U_{\alpha })}}$ is smooth for all pairs of indices α, β.