Fréchet topology on smooth vectors

Let $\pi$ be a strongly continuous unitary representation of a finite-dimensional Lie group on $H$ , and choose a real Lie-algebra basis $X_1,\ldots,X_m$ . The smooth-vector topology on $H^\infty$ is the topology generated by the norms

p_k(v)^2=\sum_{r=0}^k\ \sum_{j_1,\ldots,j_r=1}^m \|d\pi(X_{j_1})\cdots d\pi(X_{j_r})v\|_H^2, \qquad k\geq0,

where the $r=0$ term is $\|v\|_H^2$ . The operators are the derived action on smooth vectors. This topology makes $H^\infty$ a Fréchet space and is independent of the chosen basis.

Convergence and completeness

Convergence means convergence in $H$ of the vectors and of every ordered representation derivative. Equivalently, the topology is defined by $v\mapsto\|d\pi(D)v\|$ for all elements $D$ of the universal enveloping algebra, including its unit.

Completeness follows by realizing smooth vectors as the closed subspace of smooth equivariant maps in $C^\infty(G,H)$ through their orbit maps. Finite-dimensional changes of Lie-algebra basis express each word of length at most $k$ as a finite linear combination of words of the same maximal length, proving equivalence of the norm families.

Continuous infinitesimal action

The estimate

p_k(d\pi(X_j)v)\leq p_{k+1}(v)

shows that every derived operator is continuous on $H^\infty$ . By linearity this holds for every $X\in\mathfrak g$ , and compositions remain continuous. The group action on this Fréchet space is smooth.

Hilbert norm versus smooth-vector topology

The inclusion $H^\infty\hookrightarrow H$ is continuous because $p_0=\|\cdot\|_H$ . Its inherited inner product generally makes it an incomplete pre-Hilbert space; completeness above refers to the stronger topology. An operator may be continuous for the smooth-vector topology while being unbounded in the Hilbert norm.

References

Gerrit van Dijk, Karl-Hermann Neeb, Hadi Salmasian, and Christoph Zellner, On the characterization of trace class representations and Schwartz operators. §1, opening description of the smooth-vector Fréchet topology.
Karl-Hermann Neeb, On Differentiable Vectors for Representations of Infinite Dimensional Lie Groups. Theorem 4.4 and Proposition 5.4.