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 $E$ is said to be convenient or $c^\infty$-complete if whenever $c \colon \mathbb{R} \to E$ is a curve such that $l \circ c \colon \mathbb{R} \to \mathbb{R}$ is smooth for all $l \in E^*$ then $c$ is smooth.
We shall give one other characterisation which recasts the definition in language more common in functional analysis.
A locally convex topological vector space, $E$, is said to be locally complete if for $B \subseteq E$ a bounded?, closed?, absolutely convex subset then its norm space, $E_B$, is a Banach space.
The equivalence of these two forms part of Theorem 2.14 of KM.
Other equivalent characterizations of convenient vector spaces $E$ are:
$E$ is a locally convex vector space and every Mackey-Cauchy sequences? in $E$ converges.
$E$ is a locally convex vector space and for every smooth curve $c : \mathbb{R} \to E$ there is a curve $\int c$ such that $c$ is the derivative of $\int c$.
See for instance (Blute).
A Cartesian space $\mathbb{R}^n$ carries a unique structure of a convenient vector space. This also holds for Fréchet spaces in general and countable strict locally convex inductive limits thereof. (See LF space?).
For $X$ and $Y$ to convenient vector spaces, the vector space of smooth functions $C^\infty(X,Y)$ 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 category of convenient vector spaces and smooth functions between them is a cartesian closed category.
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 86, Kock-Reyes 86) and in fact already of the topos of smooth sets, hence (being concrete) of the category of diffeological spaces (Kock-Reyes 04, p. 5).
A standard textbook reference is
A survey is for instance in the slides
Results on equivalent characterizations are for instance in
The embedding of convenient vector spaces into the Cahiers topos and hence the treatment of their differential geometry by synthetic differential geometry is due to
and with a corrected definition of the site of definition in
The stronger statemet of embedding into diffeological spaces appears in