Linear Operators: General theory |
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Page 145
... uniformly on S - E . The sequence { f } converges u - uniformly to the function f if for each & > 0 there is a set E e Σ such that v ( μ , E ) < ɛ and such that { f } converges uniformly to f on S - E . It is clear that u - uniform ...
... uniformly on S - E . The sequence { f } converges u - uniformly to the function f if for each & > 0 there is a set E e Σ such that v ( μ , E ) < ɛ and such that { f } converges uniformly to f on S - E . It is clear that u - uniform ...
Page 314
... uniformly for 22 € VK . It is then clear that μ V - 1 ( u ) is a non - nega- tive element of ba ( S , E ) such that lim ( E ) = 0 uniformly for λe K. μ ( E ) -0 To prove the converse , suppose there exists a non - negative μe ba ( S , E ) ...
... uniformly for 22 € VK . It is then clear that μ V - 1 ( u ) is a non - nega- tive element of ba ( S , E ) such that lim ( E ) = 0 uniformly for λe K. μ ( E ) -0 To prove the converse , suppose there exists a non - negative μe ba ( S , E ) ...
Page 360
... uniformly for every ƒ in AC . Show that there exists a finite constant K such that for f in CBV , | ( Snf ) ( x ) | ≤K ( v ( f , [ 0 , 27 ] ) + sup f ( x ) ) , 0 ≤ x ≤ 2л . 23 Suppose that ( i ) ( Snf ) ( x ) → f ( x ) uniformly in ...
... uniformly for every ƒ in AC . Show that there exists a finite constant K such that for f in CBV , | ( Snf ) ( x ) | ≤K ( v ( f , [ 0 , 27 ] ) + sup f ( x ) ) , 0 ≤ x ≤ 2л . 23 Suppose that ( i ) ( Snf ) ( x ) → f ( x ) uniformly in ...
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