By John G. Hocking

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Also, it is clear that aP c P for every aeR+; so that P-P = LX(E; F) is a linear subspace of L~(E; F). Next, 0 ^ 8 ^ TeP=>8eP, again using the first condition in 14Ec. So L x (£; F) is solid. P If |#| ^ \TX-T2\, then 0 ^ S+ ^ \S\ < |21| + \T2\ EP, SO / S + G P . Similarly 8~eP so S = S+-S-eL*(E;F). Q x In particular, L (£; F) is a Riesz subspace of L^(£; F). (c) Finally, suppose that A is a non-empty subset of LX(E;F) and that # = sup A exists in L~(E; F). Fix ^ e ^ . Following 13Cb, set B = {sup (C U {To}) : 0 g i , C finite}.

X v xx > x. Let y be any member of A; then 0 < x\i xx — x ^ y — xeF. As F is locally order-dense, there is a z e F such that 0 < z ^ a : v ^ 1 - ^ But now seva^ ^ # + 2, soa: + 2 i s a lower bound for A in F strictly greater than x; which is impossible. X Remark The same result will be true of least upper bounds. This is sometimes expressed by saying that Fis regularly embedded in E. 17B Theorem Let E and G be Riesz spaces, and F an order-dense Riesz subspace of E. Suppose that T: F -> G is an order-continuous increasing linear map such that Ux = sup{Ty:yeF, 0 ^ y < x) exists in G for every x e E+.

S. The essential idea of 22D and 22Ga is that the polar in £~ of a solid set in £ is solid; this recurs in other contexts. 23 Fatou topologies All the locally solid topologies in which we are interested actually satisfy a further condition: 0 has a base of order-closed neighbourhoods. This leads to some much deeper results, the outstanding one being 43 23] TOPOLOGICAL RIESZ SPACES Nakano's theorem [23K], which gives a powerful sufficient condition for a topological Riesz space to be complete. Most of the section is devoted to proving this result through a series of lemmas, many of which have other applications; but as the theorem itself will not be applied in its full strength except in 64E, this work is all starred.