Convenient vector space

Remarks

Kriegl and Michor give equivalent conditions. One is that a curve $c \colon \R \to E$ is smooth as soon as $\ell \circ c$ is smooth for every continuous linear functional $\ell$. Another is that the Riemann integral $\int_0^1 c(t) \, dt$ exists in $E$ for every smooth curve $c$.

Example: every Fréchet space is convenient. A Mackey–Cauchy sequence is Cauchy, because $\mu_{n,m} B$ is eventually inside any given neighbourhood of $0$ when $B$ is bounded, and a Fréchet space is complete. More generally every sequentially complete locally convex space is convenient.

Non-example: the space $P$ of real polynomials on $[0, 1]$ with the supremum norm is not convenient. In a normed space, a sequence is Mackey–Cauchy if and only if it is Cauchy, since the unit ball is bounded. The Taylor polynomials of $e^x$ form a Cauchy sequence in $P$ whose uniform limit $e^x$ is not a polynomial, so the sequence does not converge in $P$.

A normed space is therefore convenient exactly when it is a Banach space.