and ``bra''\,\,\,\,}
&\mid\phi\rangle &&\langle F\mid
\end{array}
\label{3a}$$ A widespread example for $\Phi$ is the Schwartz space.

The vectors $\phi\in\Phi$ (in their form as kets $\mid\phi\rangle$ or bras $\langle\phi\mid$) represent physical quantities connected with the experimental apparatuses (e.g., state $\phi$ defined by a preparation apparatus or an observable $\mid\psi\rangle\langle\psi\mid$ defined by a registration apparatus (detector) fulfill $\phi,\psi\in\Phi$), the vectors $\langle F\mid$ or $\mid
F\rangle\in\Phi^\times$ represent quantities connected with the microphysical system (e.g., “scattering states” $\mid E\rangle$ or decaying states $\mid
E-i\Gamma/2\rangle$). ${\cal H}$ itself does not have any special physical meaning.

A general observable is now represented by a bounded operator $A$ in $\Phi$ (but in general by an unbounded $\overline{A}$ or $A^\dagger$ in ${\cal H}$) and corresponding to the triplet (\[3a\]) one has now a triplet of operators $$A^\dagger\left|_\Phi\right.\subset A^\dagger\subset A^\times
\label{3b}$$ In here $A^\dagger$ is the Hilbert space adjoint of $A$ (if $A$ is essentially self adjoint then $A^\dagger=\overline{A}$), $A^\dagger\left|_\Phi\right.$ denotes its restriction to the space $\Phi$, and the operator $A^\times$ in $\Phi^\times$ is the conjugate operator of $A$ defined by $$\langle A\phi\mid F\rangle=
\langle \phi\mid A^\times F\rangle\,\,\,\,
\mbox{for all $\phi\in\Phi$ and all $\mid F\rangle\in \Phi^\times$}.
\label{3c}$$ By this definition $A^\times$ is the extension of the operator $A^\dagger$ to 