Introduction to Operator Theory in Riesz Spaces
The book deals with the structure of vector lattices, i.e, Riesz spaces, and Banach lattices, as well as with operators in these spaces. The methods used are kept as simple as possible.
Almost no prior knowledge of functional analysis is required. For most applications some familiarity with the ordinary Lebesque integral is already sufficient. In this respect the book differs from other books on the subject. In most books on functional analysis (even excellent ones) Riesz spaces. Banach lattices and positive operators are mentioned only briefly, or even not at all.
The present book shows how these subjects can be treated without undue extra effort. Many of the results in the book were not yet known thirty years ago; some even were not known ten years ago.
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a-order continuous algebraic arbitrary Archimedean Archimedean Riesz space assume Banach lattice Banach space Cauchy sequence Cb(E complete Riesz space continuous functions converges in norm countable decomposition Dedekind complete Dedekind cr-complete defined definition denoted directed set disjoint complement evident example Exercise exists an element F Dedekind follows Furthermore Hence holds implies increasing sequence inequality infimum Lebesgue measure Lemma linear operator linear subspace mapping natural number non-empty subset norm bounded normed Riesz space observe order bounded operator order continuous norm order dense ordered vector space partially ordered pointwise pointwise ordering positive linear functional positive operator principal projection property projection band proof prove real numbers Riesz homomorphism Riesz subspace Riesz-Fischer property satisfying Show signed measure Similarly spectral theorem sup(u supremum u-uniform Cauchy sequence u-uniformly uniformly complete upper bound upwards directed set vector space zero