nLab convenient vector space

Idea

A convenient vector space is a locally convex topological vector space satisfying a certain completeness? property. Loosely, the completeness property is 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.

Definition

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.

Definition

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.

Definition

A locally convex topological vector space, $E$, is said to be locally complete if for $B \subseteq E$ a bounded, closed, absolutely convex subset, 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 sequence? in $E$ converges.

• $E$ is a locally convex vector space and for every smooth curve $c : \mathbb{R} \to E$ there is a curve $\int c$ (an antiderivative of $c$) such that $c$ is the derivative of $\int c$.

See for instance (Blute).

Examples

• 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 $Y$ two 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 be Cartesian closed. (See below).

Properties

• 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).

Related concepts

• convenient manifold

References

A standard textbook reference is

• Andreas Kriegl, Peter Michor: The convenient setting of global analysis, AMS (1997) (pdf)

A survey is for instance in the slides

• Richard Blute, Convenient Vector Spaces, Convenient Manifolds and Differential Linear Logic (2011) (pdf)

Results on equivalent characterizations are for instance in

• Thomas E. Gilsdorf, Strictly Webbed Convenient Locally Convex Spaces, Int. Journal of Math. Analysis, Vol. 1, 2007, no. 16, 775 - 782 (pdf)

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

• Anders Kock, Convenient vector spaces embed into the Cahiers topos, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 27 no. 1 (1986), p. 3-17 (numdam)

and with a corrected definition of the site of definition in

• Anders Kock, Gonzalo Reyes, Corrigendum and addenda to: Convenient vector spaces embed into the Cahiers topos, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 27 no. 1 (1986), p. 3-17 (numdam)

The stronger statemet of embedding into diffeological spaces appears in

• Anders Kock, Gonzalo Reyes, Categorical distribution theory; heat equation (arXiv:math/0407242)