We develop an analytical approach to quantum Gaussian states in infinite-mode representation of the Canonical Commutation Relations (CCRs), using Yosida approximations to define integrability of possibly unbounded observables with respect to a state \(\rho \) ( \(\rho \) -integrability). It turns out that all elements of the commutative \(*\) -algebra generated by a possibly unbounded \(\rho \) -integrable observable A, denoted by \(\langle A\rangle \) , are normal and \(\rho \) -integrable. Besides, \(\langle A\rangle \) can be endowed with the well-defined norm \(\|\cdot \|_\rho := {\mathrm {tr}}\left (\rho |\cdot | \right )\) . Our approach allows us to rigorously establish fundamental properties and derive key formulae for the mean value vector and the covariance operator. We additionally show that the covariance operator S of any Gaussian state is real, bounded, positive, and invertible, with the property that \(S-iJ\geq 0\) , being J the multiplication operator by \(-i\) on \(\ell _2({\mathbb N})\) .