Linear Operators: General theory |
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Page 145
... space X. It will be shown in this section that if ( S , Σ , μ ) is a measure space the various linear vector spaces of measurable and integrable functions we have encountered are complete metric spaces . Criteria for μ - measurability ...
... space X. It will be shown in this section that if ( S , Σ , μ ) is a measure space the various linear vector spaces of measurable and integrable functions we have encountered are complete metric spaces . Criteria for μ - measurability ...
Page 347
... measure or a null set . n 50 Show that no space equivalent to a space C ( S ) is equivalent to a closed subspace of a space ba ( S , Σ ) unless both are finite dimen- sional . 51 Show that no space L „ ( S , Σ , μ ) , 1 < p < ∞ , is ...
... measure or a null set . n 50 Show that no space equivalent to a space C ( S ) is equivalent to a closed subspace of a space ba ( S , Σ ) unless both are finite dimen- sional . 51 Show that no space L „ ( S , Σ , μ ) , 1 < p < ∞ , is ...
Page 405
... space s is , of course , the space of all real sequences a [ x ] as distinct from l , which is the subspace of s determined by the condition a < ∞ . The measure Mis countably additive ; the subset 2 of 8 is of μ - measure zero ; the ...
... space s is , of course , the space of all real sequences a [ x ] as distinct from l , which is the subspace of s determined by the condition a < ∞ . The measure Mis countably additive ; the subset 2 of 8 is of μ - measure zero ; the ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ