In mathematics, convenient vector spaces are locally convex vector spaces satisfying a very mild completeness condition.
Beyond Banach spaces, difficulties begin to arise; in particular, composition of continuous linear mappings stop being jointly continuous at the level of Banach spaces,[Note 1] for any compatible topology on the spaces of continuous linear mappings.
Mappings between convenient vector spaces are smooth or
This leads to a Cartesian closed category of smooth mappings between
-open subsets of convenient vector spaces (see property 6 below).
It is weaker than any other reasonable notion of differentiability, it is easy to apply, but there are smooth mappings which are not continuous (see Note 1).
It can be shown that the set of smooth curves does not depend entirely on the locally convex topology of
only on its associated bornology (system of bounded sets); see [KM], 2.11.
The final topologies with respect to the following sets of mappings into
of smooth functions with compact support on the real line, for example) it is finer than the given locally convex topology, it is not a vector space topology, since addition is no longer jointly continuous.
is said to be a convenient vector space if one of the following equivalent conditions holds (called
For maps on Fréchet spaces this notion of smoothness coincides with all other reasonable definitions.
Multilinear mappings are smooth if and only if they are bounded ([KM], 5.5).
is again a convenient vector space where the structure is given by the following injection, where
carries the topology of compact convergence in each derivative separately; see [KM], 3.11 and 3.7.
the following mapping is a linear diffeomorphism of convenient vector spaces.
This follows from the exponential law by simple categorical reasonings, see [KM], 3.13.
Convenient calculus of smooth mappings appeared for the first time in [Frölicher, 1981], [Kriegl 1982, 1983].
Convenient calculus (having properties 6 and 7) exists also for: The corresponding notion of convenient vector space is the same (for their underlying real vector space in the complex case) for all these theories.
The exponential law 6 of convenient calculus allows for very simple proofs of the basic facts about manifolds of mappings.
is described in the following diagram: It induces an atlas of charts on the space
Trivializing the pull back vector bundle
and applying the exponential law 6 leads to the diffeomorphism All chart change mappings are smooth (
is a smooth manifold modeled on Fréchet spaces.
As a consequence of the chart structure, the tangent bundle of the manifold of mappings is given by Let
Multiplication and inversion are denoted by: The notion of a regular Lie group is originally due to Omori et al. for Fréchet Lie groups, was weakened and made more transparent by J. Milnor, and was then carried over to convenient Lie groups; see [KM], 38.4.
is called regular if the following two conditions hold: If
induces the evolution operator via which satisfies the ordinary differential equation Given a smooth curve in the Lie algebra,
, then the solution of the ordinary differential equation depends smoothly also on the further variable
QED An overview of applications using geometry of shape spaces and diffeomorphism groups can be found in [Bauer, Bruveris, Michor, 2014].