Nuclear space

Remarks

Example: the Schwartz space $\mathcal{S}(\R^n)$, the space $C^\infty(\Omega)$ of smooth functions on an open set $\Omega \subseteq \R^n$, and the space $H(\Omega)$ of holomorphic functions on an open set $\Omega \subseteq \C$ are nuclear. Non-example: an infinite-dimensional normed space $E$ is not nuclear. Take $p$ to be the norm; then every continuous seminorm $q \ge p$ is equivalent to the norm. So $\iota_{q, p}$ is an isomorphism of infinite-dimensional Banach spaces, which is not compact and hence not nuclear. Grothendieck's original definition is equivalent: $E$ is nuclear if and only if, for every Banach space $F$, the projective and injective tensor topologies on $E \otimes F$ agree. Every nuclear Fréchet space is a Montel space.