By Constantin Banica, Octavian Stanasila
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Extra info for Algebraic Methods in the Global Theory of Complex Spaces
Make use of GT, V, §3, Exer. ) d) Give an example of a Riesz space not reduced to 0 that contains no maximal isolated subspace (cf. §1, Exer. 5 b». ~ 6) Let E be a Riesz space, 0 the fully lattice-ordered space of relatively bounded linear forms on E, and F a co-lattice subspace of 0 (§1, Exer. 3). For every x E E, the mapping x' f--t (x, x') of F into R is a relatively bounded linear form Ux on F, and the mapping x f--t Ux is an increasing linear mapping of E into the fully latticeordered space 0' of relatively bounded linear forms on F.
If Bi and B2 are two supplementary bands in E, show that H is the ordered direct sum of Hi = H n Bi and H2 = H n B2 , and that the Riesz space E/H is the ordered direct sum of the Riesz spaces Bi/Hi and B2/H2. ) be a family of fully lattice-ordered spaces, E the product space of the E~ (which is fully lattice-ordered). Show that if B is a band in E, then each of its projections B~ = pr~(B) is a band in E~, and B is identical to the product of the B~. From this, deduce the determination of the bands in the space RA of mappings of a set A into R.
Lemma 2. - Let X be a locally compact space, K a compact subset of X, E a locally convex space, q a continuous semi-norm on E, and
0, there exists a continuous partition of unity ('Pj )O:::;j:::;n on X having the following properties: (i) Supp( 'Pj) c K for 1 ~ j ~ n. (ii) If, for 1 ~ j ~ n, Xj is any point of Supp('Pj), then, for every function f E
Algebraic Methods in the Global Theory of Complex Spaces by Constantin Banica, Octavian Stanasila