Convenient vector space

In mathematics, convenient vector spaces are locally convex vector spaces satisfying a very mild completeness condition.

Traditional differential calculus is effective in the analysis of finite-dimensional vector spaces and for Banach spaces. Beyond Banach spaces, difficulties begin to arise; in particular, composition of continuous linear mappings stop being jointly continuous at the level of Banach spaces, for any compatible topology on the spaces of continuous linear mappings.

Mappings between convenient vector spaces are smooth or ${\displaystyle C^{\infty }}$ if they map smooth curves to smooth curves. This leads to a Cartesian closed category of smooth mappings between ${\displaystyle c^{\infty }}$-open subsets of convenient vector spaces (see property 6 below). The corresponding calculus of smooth mappings is called convenient calculus. 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). This type of calculus alone is not useful in solving equations.

Let ${\displaystyle E}$ be a locally convex vector space. A curve ${\displaystyle c:\mathbb {R} \to E}$ is called smooth or ${\displaystyle C^{\infty }}$ if all derivatives exist and are continuous. Let ${\displaystyle C^{\infty }(\mathbb {R} ,E)}$ be the space of smooth curves. It can be shown that the set of smooth curves does not depend entirely on the locally convex topology of ${\displaystyle E,}$ 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 ${\displaystyle E}$ coincide; see [KM], 2.13.

This topology is called the ${\displaystyle c^{\infty }}$-topology on ${\displaystyle E}$ and we write ${\displaystyle c^{\infty }E}$ for the resulting topological space. In general (on the space ${\displaystyle D}$ 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. Namely, even ${\displaystyle c^{\infty }(D\times D)\neq \left(c^{\infty }D\right)\times \left(c^{\infty }D\right).}$ The finest among all locally convex topologies on ${\displaystyle E}$ which are coarser than ${\displaystyle c^{\infty }E}$ is the bornologification of the given locally convex topology. If ${\displaystyle E}$ is a Fréchet space, then ${\displaystyle c^{\infty }E=E.}$

A locally convex vector space ${\displaystyle E}$ is said to be a convenient vector space if one of the following equivalent conditions holds (called ${\displaystyle c^{\infty }}$-completeness); see [KM], 2.14.

Here a mapping ${\displaystyle f:\mathbb {R} \to E}$ is called ${\displaystyle {\text{Lip}}^{k}}$ if all derivatives up to order ${\displaystyle k}$ exist and are Lipschitz, locally on ${\displaystyle \mathbb {R} }$.

Let ${\displaystyle E}$ and ${\displaystyle F}$ be convenient vector spaces, and let ${\displaystyle U\subseteq E}$ be ${\displaystyle c^{\infty }}$-open. A mapping ${\displaystyle f:U\to F}$ is called smooth or ${\displaystyle C^{\infty }}$, if the composition ${\displaystyle f\circ c\in C^{\infty }(\mathbb {R} ,F)}$ for all ${\displaystyle c\in C^{\infty }(\mathbb {R} ,U)}$. See [KM], 3.11.