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Page 68
... limit . PROOF . If x and y are both weak limits of a generalized sequence , then for each ** € X * , x * x x * y , x * ( x - y ) = 0 and x = Corollary 14. Q.E.D. = y , by 27 LEMMA . A weakly convergent sequence { x } of points in a ...
... limit . PROOF . If x and y are both weak limits of a generalized sequence , then for each ** € X * , x * x x * y , x * ( x - y ) = 0 and x = Corollary 14. Q.E.D. = y , by 27 LEMMA . A weakly convergent sequence { x } of points in a ...
Page 126
... limit inferior and the limit superior of { E } by the equations lim inf En - n En 00 00 UnEm , n = 1 m = n lim sup En - n 00 00 nu Em · n = 1 m = n If lim inf , E1 = lim sup , En , { E } is said to be convergent , and we write the ...
... limit inferior and the limit superior of { E } by the equations lim inf En - n En 00 00 UnEm , n = 1 m = n lim sup En - n 00 00 nu Em · n = 1 m = n If lim inf , E1 = lim sup , En , { E } is said to be convergent , and we write the ...
Page 658
... limit . ( iii ) T lim f ( pt ( x ) ) dt , T which , in physical theories , is assumed to exist . Thus , the mathematician is led to the problem of determining whether or not the limit ( iii ) exists . The next four sections are devoted ...
... limit . ( iii ) T lim f ( pt ( x ) ) dt , T which , in physical theories , is assumed to exist . Thus , the mathematician is led to the problem of determining whether or not the limit ( iii ) exists . The next four sections are devoted ...
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