By H. G. Dales, F.K. Dashiell Jr., A.T.-M. Lau, D. Strauss
This booklet provides a coherent account of the idea of Banach areas and Banach lattices, utilizing the areas C_0(K) of continuing capabilities on a in the neighborhood compact area ok because the major example. The research of C_0(K) has been a big quarter of useful research for lots of years. It supplies numerous new buildings, a few regarding Boolean jewelry, of this space as good as many effects at the Stonean house of Boolean rings. The book also discusses while Banach areas of continuing features are twin areas and once they are bidual spaces.
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Extra resources for Banach Spaces of Continuous Functions as Dual Spaces (CMS Books in Mathematics)
Since π is continuous, L 0 is a compact subspace of L. / say U = intL L 0 , and then set Assume towards a contradiction that intL L 0 = 0, K1 = π −1 (L \ U), so that K0 ∪ K1 is a closed subspace of K with π (K0 ∪ K1 ) = L. / a Since π is irreducible, K0 ∪ K1 = K, and so π −1 (U) ⊂ K0 , whence intK K0 = 0, contradiction. Thus L 0 ∈ KL . 23. Let K be a non-empty, compact, metric space without any isolated points. Then there is an irreducible surjection from the Cantor set Δ onto K. Proof. 6(i), there is a continuous surjection η : Δ → K.
12. Let B be a Boolean ring. A measure on B is a map μ : B → C such that μ (a ∨ b) = μ (a) + μ (b) whenever a, b ∈ B with a ∧ b = 0. A measure μ on B is: positive if μ (a) ≥ 0 (a ∈ B); normal if limα μ (aα ) = 0 for each net (aα ) in 0; and σ -normal if limn→∞ μ (an ) = 0 for each sequence (an ) in B B such that aα such that an 0. 27. 7 Boolean algebras and Boolean rings 43 For μ , ν ∈ N R (B), set μ ≤ ν if μ (b) ≤ ν (b) (b ∈ B).
14, below. 16, on page 45, we shall exhibit a space G I , the Gleason cover of the unit interval, and note that G I is an infinite, separable, Stonean space without isolated points, and hence G I is not homeomorphic to either β N or N∗ . One might wonder if any two infinite, separable Stonean spaces without isolated points are homeomorphic. 16. 25], for example. 11. Let X be a completely regular space. Then X is an F-space when every finitely generated ideal in the algebra C R (X) is principal.