Linear Operators: General theory |
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Page vi
... present work is written for the student as well as for the mature mathematician . Much of the text has grown directly out of lec- tures given by the authors over many years , and the two parts are de- signed to form suitable texts for a ...
... present work is written for the student as well as for the mature mathematician . Much of the text has grown directly out of lec- tures given by the authors over many years , and the two parts are de- signed to form suitable texts for a ...
Page 285
... present theorem is a corollary of Theorem 6.18 . Q.E.D. 8. The Spaces L1 ( S , Σ , μ ) The spaces L , ( S , Σ , μ ) , 1 ≤ p < ∞ , have already been studied in Chapter III . In particular it was shown in Theorem III.6.6 that they are B ...
... present theorem is a corollary of Theorem 6.18 . Q.E.D. 8. The Spaces L1 ( S , Σ , μ ) The spaces L , ( S , Σ , μ ) , 1 ≤ p < ∞ , have already been studied in Chapter III . In particular it was shown in Theorem III.6.6 that they are B ...
Page 286
... present , that u ( S ) < ∞o . If XE is the characteristic function of the set E € Σ , then , if { E } is a disjoint sequence of measurable subsets of S and U1 E ; = E。, -1 % E Eo , it follows from III.6.16 that 1 XE , = XE , the ...
... present , that u ( S ) < ∞o . If XE is the characteristic function of the set E € Σ , then , if { E } is a disjoint sequence of measurable subsets of S and U1 E ; = E。, -1 % E Eo , it follows from III.6.16 that 1 XE , = XE , the ...
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