Linear Operators: General theory |
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Page 182
... clear that u1 E it is clear that μ1 is well defined on 1. Since ( EF ) = 6 - 1 ( E ) 6 - 1 ( F ) , 6 - 1 ( E ' ) = { 4 - 1 ( E ) } ' , and -1 ( UE ; ) = 1-1 ( E ) , it is clear that is a field and that = 1 is additive on 1 . These ...
... clear that u1 E it is clear that μ1 is well defined on 1. Since ( EF ) = 6 - 1 ( E ) 6 - 1 ( F ) , 6 - 1 ( E ' ) = { 4 - 1 ( E ) } ' , and -1 ( UE ; ) = 1-1 ( E ) , it is clear that is a field and that = 1 is additive on 1 . These ...
Page 282
... clear that T ( ɛ ) ≤ T ( 8 ) if ɛ < 8 and that —t e T ( ɛ ) whenever t e T ( E ) . The function f is said to be almost periodic if it is continuous and if for every ɛ > 0 there is an L = L ( e ) > 0 such that every interval in R of ...
... clear that T ( ɛ ) ≤ T ( 8 ) if ɛ < 8 and that —t e T ( ɛ ) whenever t e T ( E ) . The function f is said to be almost periodic if it is continuous and if for every ɛ > 0 there is an L = L ( e ) > 0 such that every interval in R of ...
Page 292
... clear that if F1 and F2 are elements of Σ3 , then F1F2 € 23. It is also clear that if F1 € Σ , then S - F1 € 23 , and that if F1 , F2 € 23 with FF2 = 6 , then F1 UF2 € 23 . F2 It follows that Σ is a field . If { F } is a sequence of ...
... clear that if F1 and F2 are elements of Σ3 , then F1F2 € 23. It is also clear that if F1 € Σ , then S - F1 € 23 , and that if F1 , F2 € 23 with FF2 = 6 , then F1 UF2 € 23 . F2 It follows that Σ is a field . If { F } is a sequence of ...
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