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Page 177
... assumed that v ( 2 , F ) < ∞ . For , if ƒ is integrable then v ( 2 , F ) < ∞o for all Fe 2 ; if f is only assumed to be positive and measurable we can put F { s \ s € F , f ( s ) ≤n } . Then - F = F and v ( λ , F „ ) < ∞ for each n ...
... assumed that v ( 2 , F ) < ∞ . For , if ƒ is integrable then v ( 2 , F ) < ∞o for all Fe 2 ; if f is only assumed to be positive and measurable we can put F { s \ s € F , f ( s ) ≤n } . Then - F = F and v ( λ , F „ ) < ∞ for each n ...
Page 650
... assumed that p , ( 2 ) f ( 2 ) ≤ M in a strip R ( 2 ) | < w + ε , where ε is independent of n . In the first paragraph of the proof of Lemma 7 it was seen that | R ( 2 , A ) | is uniformly bounded in any half plane R ( 2 ) > + ε and in ...
... assumed that p , ( 2 ) f ( 2 ) ≤ M in a strip R ( 2 ) | < w + ε , where ε is independent of n . In the first paragraph of the proof of Lemma 7 it was seen that | R ( 2 , A ) | is uniformly bounded in any half plane R ( 2 ) > + ε and in ...
Page 657
... assumed to be governed by the classical Hamiltonian equations and so is subject to a principle of scientific determinism whereby it is known that an initial state a will , after t seconds have elapsed , have passed into a uniquely ...
... assumed to be governed by the classical Hamiltonian equations and so is subject to a principle of scientific determinism whereby it is known that an initial state a will , after t seconds have elapsed , have passed into a uniquely ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
Copyright | |
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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