the space $\Phi^\times$ (and not the extension of the operator $A$ which is most often used in mathematics). A very important point is that the operator $A^\times$ is only defined for an operator $A$ which is continuous=bounded in $\Phi$, then $A^\times$ is a continuous (but not bounded) operator in $\Phi^\times$. It is impossible in quantum mechanics (empirically) to restrict oneself to continuous=bounded operators $\overline{A}$ in ${\cal H}$, but one can restrict oneself to algebras of observables $\{A,B\ldots\}$ described by continuous operators in $\Phi$. Then $A^\times,B^\times\ldots$ are defined and continuous in $\Phi^\times$. If $A$ in (\[3c\]) is not self-adjoint then $A^\dagger\mid_{\Phi}$ need not be a continuous operator in $\Phi$ even if $A$ is, but one can still define the conjugate $A^\times$ which is continuous in $\Phi^\times$.

A generalized eigenvector $F\in\Phi^\times$ of an operator $A$ is defined by $$\langle A\phi\mid F\rangle=
\langle \phi\mid A^\times F\rangle=
\omega\langle \phi\mid F\rangle\,\,\,
\mbox{for all $\phi\in\Phi$}
\label{3d}$$ where the complex number $\omega$ is called the generalized eigenvalue. This is also written as $$A^\times\mid F\rangle=
\omega\mid F\rangle.
\label{3e}$$ For an essentally self-adjoint operator $A^\dagger=\overline{A}$ (= closure of $A$) this is often also written as $$A\mid F\rangle=
\omega\mid F\rangle
\label{3en}$$ especially if one suppresses the mathematical subtleties and acts 