EXISTENCE OF BADE FUNCTIONALS FOR COMPLETE BOOLEAN-ALGEBRAS OF PROJECTIONS IN FRECHET SPACES

A classical result of W. Bade states that if M is any sigma-complete B oolean algebra of projections in an arbitrary Banach space X then, for every x(0) is an element of X, there exists an element x' (called a B ade functional for x(0) with respect to M) in the dual space X', with the following two properties: (i) M bar right arrow [Mx(0), x'] is non -negative on M and, (ii) Mx(0) = 0 whenever M is an element of M satis fies [Mx(0), x') = 0. It is shown that a Frechet space X has this prop erty if and only ii it does not contain an isomorphic copy of the sequ ence space omega = C-N.