Let ${\mathscr{H}}_{\mu }$ be the Zemanian space of Hankel transformable functions, and let ${\mathscr{H}}_{\mu }^{\text{'}}$ be its dual space. In this paper ${\mathscr{H}}_{\mu }$ is shown to be nuclear, hence Schwartz, Montel and reflexive. The space $\text{O}$, also introduced by Zemanian, is completely characterized as the set of multipliers of ${\mathscr{H}}_{\mu }$ and of ${\mathscr{H}}_{\mu }^{\text{'}}$. Certain topologies are considered on $𝒪$, and continuity properties of the multiplication operation with respect to those topologies are discussed.