  We consider a category of continuous Hilbert space representations and a
category of smooth Frechet representations, of a real Jacobi group $G$. By
Mackey's theory, they are respectively equivalent to certain categories of
representations of a real reductive group $\widetilde L$. Within these
categories, we show that the two functors of taking smooth vectors for $G$, and
for $\widetilde L$, are consistent with each other. By using
Casselman-Wallach's theory of smooth representations of real reductive groups,
we define matrix coefficients for distributional vectors of certain
representations of $G$. We also formulate Gelfand-Kazhdan criteria for Jacobi
groups which could be used to prove the multiplicity one theorem for
Fourier-Jacobi models.
