case, because on the level of the mathematical rigor employed by the physicist the RHS formulation of quantum mechanics is like Dirac’s bra- and ket-formalism. When physicists talk about the Hilbert space they mostly mean a pre-Hilbert space, i.e., a linear space $\Psi$ with a “scalar product”, denoted by $(\psi,F)$ or $\langle\psi\mid F\rangle$ without worrying about its topological completion. The Hilbert space of mathematicians is a much more complicated structure, its elements being represented not by functions but by classes of functions whose elements differ on a set of Lebesgue measure zero, a mathematically complicated and physically useless concept (because the apparatus resolution is described by smooth functions). The RHS is the same linear space $\Psi$ only with different topological completions: one completes $\Psi$ with respect to a topology that is stronger than the topology given by the ${\cal H}$-space norm (e.g., one uses a countable number of norms) to obtain the space $\Phi\subset{\cal H}$ and considers in addition the topological dual to $\Phi$, i.e., the space of [*continuous*]{} antilinear functionals of $\Phi$ denoted by $\Phi^\times$. Then one obtains the triplet $$\begin{array}{rccc}
\mbox{Gelfand triplet: }\Psi\subset
&\Phi&\subset{\cal H}={\cal H}^\times\subset &\Phi^\times\\
&\in &&\in\\
\mbox{with elements ``bra'' and ``ket''\,\,\,\,}&\langle\phi\mid
&&\mid F\rangle\\
\mbox{or ``ket'' 