Linear Operators: General theory |
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Page 287
... evident from the Hölder inequality that any g e L , deter- mines an a * L * satisfying ( i ) , so that the mapping a * →→→ g is a one - to - one isometric map of L * onto L. Since the linearity of the map is evident , the theorem is ...
... evident from the Hölder inequality that any g e L , deter- mines an a * L * satisfying ( i ) , so that the mapping a * →→→ g is a one - to - one isometric map of L * onto L. Since the linearity of the map is evident , the theorem is ...
Page 299
... evident that + ∞0 lim x ( x + y ) —x ( y ) | P dy 0 < x 81 = 0 if χ is the characteristic function of a finite interval . Thus lim f ± g , ( x + y ) — g ; ( y ) | P dy = 0 for each function g ;; and hence x → 0 lim sup 04x + ∞ [ + ...
... evident that + ∞0 lim x ( x + y ) —x ( y ) | P dy 0 < x 81 = 0 if χ is the characteristic function of a finite interval . Thus lim f ± g , ( x + y ) — g ; ( y ) | P dy = 0 for each function g ;; and hence x → 0 lim sup 04x + ∞ [ + ...
Page 337
... evident that BV ( I ) = BV ( I ) N. Thus BV ( I ) is isometrically isomorphic to the direct sum of ba ( I , Σ ) and a one - dimensional space . From this , the following theorem is evident ( cf. 9.9 ) . 1 THEOREM . The space BV ( I ) is ...
... evident that BV ( I ) = BV ( I ) N. Thus BV ( I ) is isometrically isomorphic to the direct sum of ba ( I , Σ ) and a one - dimensional space . From this , the following theorem is evident ( cf. 9.9 ) . 1 THEOREM . The space BV ( I ) is ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ