Chapter 35: Riesz spaces

351
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.

352
Riesz 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.

353
Archimedean and Dedekind complete Riesz spaces

Order-dense subspaces; bands; Dedekind (σ)-complete spaces;
order-closed Riesz subspaces; order units; *f*-algebras.

354
Banach lattices

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.

355
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.

356
Dual spaces

The spaces *U*^{~}, *U*^{×}, *U**; biduals, embeddings
*U*→*V*^{×} where *V*Í*U*^{~}; 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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Revised 17.2.2017