Let X be the domain of a linear
transformation A. Certain subspaces of the second algebraic conjugate Xff, obtained
by the application of a weak completion process to some suitable subspace of X,
may be regarded as spaces of generalized elements to which A has a natural
extension. When A is a closed Hilbert space transformation, its domain can
in this way be extended to a weakly complete space (Theorem 1). For a
selfadjoint operator T this extension X may be regarded as the dual of a perfect
countably Hilbert space precisely if T has a compact inverse (Theorem 2).
Any element in X is obtained by a repeated application of the extended
transformation T to some element in X (Theorem 3). A discussion of the
extension of functions of T to X, and a spectral theory for T conclude the
paper.
