Linear Operators: General theory |
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Page 287
... evident from the Hölder inequality that any g e La deter- mines an x * 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 ...
... evident from the Hölder inequality that any g e La deter- mines an x * 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 ...
Page 299
... evident that + ∞ lim \ x ( x + y ) —x ( y ) " dy 0 -∞ x0 if X is the characteristic function of a finite interval . Thus lim f ± g , ( x + y ) — g ; ( y ) | o dy - 0 for each function g ;; and hence · + ∞ lim sup ( ** \ f ( x + y ) ...
... evident that + ∞ lim \ x ( x + y ) —x ( y ) " dy 0 -∞ x0 if X is the characteristic function of a finite interval . Thus lim f ± g , ( x + y ) — g ; ( y ) | o dy - 0 for each function g ;; and hence · + ∞ lim sup ( ** \ f ( x + y ) ...
Page 337
... evident that _BV ( I ) = BV1 ( 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 0 THEOREM . The space BV ...
... evident that _BV ( I ) = BV1 ( 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 0 THEOREM . The space BV ...
Contents
Preliminary Concepts | 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 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 ergodic exists f₁ finite dimensional function defined function f 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-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ