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 $\left\{U_{\alpha }\right\}_{\alpha \in I},$ and a collection of homeomorphisms $\phi _{\alpha }:U_{\alpha }\to F_{\alpha }$ onto their images, where $F_{\alpha }$ are Fréchet spaces, such that

\phi _{\alpha \beta }:=\phi _{\alpha }\circ \phi _{\beta }^{-1}|_{\phi _{\beta }\left(U_{\beta }\cap U_{\alpha }\right)}

is smooth for all pairs of indices $\alpha ,\beta .$

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