Chapter 35: Riesz spaces
Partially ordered linear spaces
Partially ordered linear spaces; positive cones; suprema and infima; positive linear operators; order-continuous linear operators; Riesz homomorphisms; quotient spaces; reduced powers; representation of p.o.l.ss as subspaces of reduced powers of R; Archimedean spaces.
Riesz spaces; identities; general distributive laws; Riesz homomorphisms; Riesz subspaces; order-dense subspaces and order-continuous operators; bands; the algebra of complemented bands; the algebra of projection bands; principal bands; f-algebras.
Archimedean and Dedekind complete Riesz spaces
Order-dense subspaces; bands; Dedekind (σ)-complete spaces; order-closed Riesz subspaces; order units; f-algebras.
Riesz norms; Fatou norms; the Levi property; order-continuous norms; order-unit norms; M-spaces; are isomorphic to C(X) for compact Hausdorff X; L-spaces; uniform integrability in L-spaces.
Spaces of linear operators
Order-bounded linear operators; the space L~(U;V); order-continuous operators; extension of order-continuous operators; the space L×(U,V); order-continuous norms.
The spaces U~, U×, U*; biduals, embeddings U→V× where VÍU~; perfect Riesz spaces; L- and M-spaces; uniformly integrable sets in the dual of an M-space; relative weak compactness in L-spaces.
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