nLab convenient vector space

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Contents

Contents

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 EE is said to be convenient or c c^\infty-complete if whenever c:Ec \colon \mathbb{R} \to E is a curve such that lc:l \circ c \colon \mathbb{R} \to \mathbb{R} is smooth for all lE *l \in E^* then cc 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, EE, is said to be locally complete if for BEB \subseteq E a bounded, closed, absolutely convex subset, its norm space?, E BE_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 EE are:

See for instance (Blute).

Examples

  • A Cartesian space n\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 XX and YY two convenient vector spaces, the vector space of smooth functions C (X,Y)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

References

A standard textbook reference is

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

Last revised on July 4, 2023 at 16:47:11. See the history of this page for a list of all contributions to it.