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Page 100
... everywhere , or , if u is understood , simply almost everywhere , or for almost all s in S , if it is true except for those points s in a μ - null set . The phrase " almost everywhere " is customarily abbreviated " a.e. " Thus , if lim ...
... everywhere , or , if u is understood , simply almost everywhere , or for almost all s in S , if it is true except for those points s in a μ - null set . The phrase " almost everywhere " is customarily abbreviated " a.e. " Thus , if lim ...
Page 150
... everywhere . ( b ) If ( S , Σ , μ ) is a finite measure space , an almost everywhere convergent sequence of measurable functions is convergent in u - measure . 14 COROLLARY . If ( S , E , μ ) is a measure space , and { fn } is a ...
... everywhere . ( b ) If ( S , Σ , μ ) is a finite measure space , an almost everywhere convergent sequence of measurable functions is convergent in u - measure . 14 COROLLARY . If ( S , E , μ ) is a measure space , and { fn } is a ...
Page 722
... everywhere for fe L ,, 1 ≤ p < ∞ . Show that if S is a topological space , and if , for each 2 > 0 , R ( 2 ; A ) ƒ ... everywhere . 22 Let T be a transformation in L1 , T≥0 , | T1 ≤ 1 , and A ( n ) = A ( T , n ) . Show that if f is ...
... everywhere for fe L ,, 1 ≤ p < ∞ . Show that if S is a topological space , and if , for each 2 > 0 , R ( 2 ; A ) ƒ ... everywhere . 22 Let T be a transformation in L1 , T≥0 , | T1 ≤ 1 , and A ( n ) = A ( T , n ) . Show that if f is ...
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