A convenient vector spaces is a locally convex topological vector space satisfying a certain completeness? property. Loosely, the completeness property is that such that “all derivatives which ought to exist actually do”.
The term convenient vector space was introduced by Kriegl and Michor as part of their theory of global analysis. It is equivalent to the notion locally complete which is more usual in functional analysis.
There are various equivalent definitions of a convenient vector space, a long list is given in Theorem 2.14 of KM. We shall give the definition that is closest to its raison d’être, namely the existence of derivatives.
A locally convex topological vector space is said to be convenient or -complete if whenever is a curve such that is smooth for all then is smooth.
We shall give one other characterisation which recasts the definition in language more common in functional analysis.
The equivalence of these two forms part of Theorem 2.14 of KM.
Other equivalent characterizations of convenient vector spaces are:
See for instance (Blute).
A Cartesian space carries a unique structure of a convenient vector space.
For and to convenient vector spaces, the vector space of smooth functions is again a convenient vector space. This is to a large degree the motivating example. It makes the category of convenient vector spaces by Cartesian closed. (See below).
The definition of a convenient vector space has a natural interpretation in terms of Frölicher spaces. It may also be related to synthetic differential geometry: convenient vector spaces form a full subcategory of the Cahiers topos (Kock,KockReyes).
A standard textbook reference is
A survey is for instance in the slides
Results on equivalent characterizations are for instance in
and with a corrected definition of the site of definition in