  In this paper we show that a result of Gross and Kuelbs, used to study
Gaussian measures on Banach spaces, makes it possible to construct an adjoint
for operators on separable Banach spaces. This result is used to extend well
known theorems of von Neumann and Lax. We also partially solve an open problem
on the existence of a Markushevich basis with unit norm and prove that all
closed densely defined linear operators on a separable Banach space can be
approximated by bounded operators. This last result extends a theorem of
Kaufman for Hilbert spaces and allows us to define a new metric for closed
densely defined linear operators on Banach spaces. As an application, we obtain
a generalization of the Yosida approximator for semigroups of operators.
