Linear Operators: General theory |
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Page 287
... evident from the Hölder inequality that any ge L , deter- mines an * 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 proved ...
... evident from the Hölder inequality that any ge L , deter- mines an * 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 proved ...
Page 299
... evident that + ∞ ( ** \ x ( x + y ) —x ( y ) | 3dy = 0 81 if x is the characteristic function of a finite interval . Thus lim fg , ( x + y ) —g , ( y ) dy = 0 for each function g1 ; and hence 04 - x • + ∞ | o + ∞ lim sup ( *** \ f ...
... evident that + ∞ ( ** \ x ( x + y ) —x ( y ) | 3dy = 0 81 if x is the characteristic function of a finite interval . Thus lim fg , ( x + y ) —g , ( y ) dy = 0 for each function g1 ; and hence 04 - x • + ∞ | o + ∞ lim sup ( *** \ f ...
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 ...
... 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 ...
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 ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f g₁ Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ