A convenient vector space is a locally convex topological vector space that satisfies a specific completeness condition regarding smooth curves, making it a suitable setting for differential calculus in infinite dimensions. The concept was systematically developed by Andreas Kriegl and Peter Michor in their 1997 monograph The Convenient Setting of Global Analysis as a foundation for the convenient calculus (also known as $c^\infty$-calculus).
Definition
Let $E$ be a locally convex vector space. A curve $c: \mathbb{R} \to E$ is called smooth ($c^\infty$) if all derivatives $c^{(n)}(t)$ exist in the topology of $E$. The space $E$ is called a convenient vector space if one of the following equivalent conditions holds:
- $c^\infty$-completeness: For every smooth curve $c: \mathbb{R} \to E$, there exists a smooth curve $d: \mathbb{R} \to E$ such that $d' = c$ (i.e., every smooth curve has an antiderivative).
- Mackey completeness: Every Mackey-Cauchy sequence (or net) converges in $E$. A sequence $(x_n)$ is Mackey-Cauchy if there exists a bounded, absolutely convex set $B \subset E$ and a sequence of scalars $(\mu_n)$ converging to 0 such that $x_n - x_m \in \mu_n B$ for all $n,m$.
- $c^\infty$-topology: The $c^\infty$-topology (the final topology with respect to all smooth curves) coincides with the original locally convex topology.
Properties
- Completeness: Every convenient vector space is sequentially complete. However, it need not be complete in the standard sense of Cauchy nets/filters (e.g., the space $\mathbb{R}^\mathbb{N}$ with the product topology is convenient but not complete).
- Closure properties: The category of convenient vector spaces is closed under limits, colimits, and the formation of spaces of smooth mappings $C^\infty(M, E)$ for finite-dimensional manifolds $M$.
- Bornology: The structure of a convenient vector space is determined by its bornology (system of bounded sets). A linear map between convenient vector spaces is smooth if and only if it is bornological (maps bounded sets to bounded sets).
Significance
Convenient vector spaces form the objects of the convenient category (often denoted $\mathcal{C}^\infty$ or $\text{Con}$), which is a Cartesian closed category. This categorical property is the primary motivation for the definition: it ensures that the space of smooth maps $C^\infty(E, F)$ between two convenient vector spaces is again a convenient vector space, and the exponential law holds: $$ C^\infty(E \times F, G) \cong C^\infty(E, C^\infty(F, G)) $$ This allows for a rigorous treatment of infinite-dimensional manifolds and Lie groups (e.g., diffeomorphism groups, loop groups) using standard manifold theory modeled on convenient vector spaces rather than Banach spaces.
Examples
- All Fréchet spaces (and hence all Banach spaces) are convenient vector spaces.
- The space $\mathbb{R}^\mathbb{N}$ (the space of all real sequences with the product topology) is convenient but not metrizable.
- The space $\mathcal{D}(\Omega)$ of test functions (compactly supported smooth functions) on an open domain $\Omega \subset \mathbb{R}^n$ with its standard strict inductive limit topology (LF-space) is convenient.
- The space $\mathcal{E}(\Omega) = C^\infty(\Omega)$ with the topology of uniform convergence of all derivatives on compact sets is a Fréchet space, hence convenient.
- The space of distributions $\mathcal{D}'(\Omega)$ with the weak-* topology is convenient.
Relation to Other Structures
- Keller's $C^\infty_c$-theory: Convenient calculus subsumes and generalizes earlier approaches to calculus on Fréchet spaces (Keller, 1974).
- Bastiani calculus: On Fréchet spaces, convenient smoothness coincides with the Bastiani (or Keller) definition of smooth maps.
- Diffeological spaces: Every convenient vector space carries a natural diffeology (where plots are smooth curves), making the category of convenient vector spaces a full subcategory of the category of diffeological spaces.
