# Complemented subspaces of A, H1, and H by Casazza P. G. By Casazza P. G.

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15) Proposition. Let (I,D) be a P-diagram. [: DS-----~Dsi i is an epimorphism. Proof. S. is ~. g. Evidently every diagram is the limit of representable diagrams and an inverse limit of monomorphisms is a monomorphism. 57 4. The Lubkin completion process. 1) In this section we show how to "complete" a given diagram to a P-diagram. This construction was first described b y Lubkin in his original proof of the abelian category imbedding, [Lu]. As a matter of fact, Lubkin observed then that there was nothing inherently abelian in his proof.

Y w h i c h far q u o t i e n t s . Proof. )D ... is a m o n o m o r p h i s m , and if we use the c o m p o s i t e comes >X be a small diagram. It is a set since each G has o n l y a set of m a p s (X,D) , t h e n (X,D). pair, (X~D) . diagram. in b y its k e r n e l of X x X. 3) set of X is d e t e r m i n e d quotients. through (X,D) X, it be- property. regular and such that each o b j e c t c a t e g o r y w i t h a set of has o n l y a set o f Then X is complete. For a d i a g r a m D: I >X, a limit of D is a t e r m i n a l object of (X,D).

S are equal. is an i s o m o r p h i s m . Let the k e r n e l Proposition. object Let ~T where S is a p a r t i a l terminal pair. f: X ~ S be c o n s t a n t . T h e n S is a p a r t i a l and S = supp X. As any c o n s t a n t ~ supp X f: S f is an i s o m o r p h i s m . Consider terminal Proof. pl,P2: >S Trivial. object. Proof. object. pl,P2 ~ S × S projections d- T h e d i a g o n a l Proof. 5). 5) . 7) Let S u p p X denote the full subcategory the partial terminal objects. 8) Proposition.