It has been proved that if an involution \({}^+\) exists in a subring of continuous linear maps in a locally convex topological vector space \({\mathcal S}\) with the property \((\xi A)^+=\xi^*A^+\), and \(A^+A=0\) iff \(A=0\), then there exists in \({\mathcal S}^ a \)Hermitian scalar product \(<, >\) such that \({}^+\) is the adjoint operation with respect to \(<, >\). the existence of a corresponding Gel’fand triple \({\mathcal S}\subset H\subset {\mathcal S}'\) has been shown.