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 L ( S , E , 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 L ( S , E , 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 μ ( S ) < ∞ . If XE is the characteristic function of the set E € 2 , then , if { E } is a disjoint sequence of measurable subsets of S and UE , Eo , it follows from III.6.16 that the series converging in the norm of L ...
... present , that μ ( S ) < ∞ . If XE is the characteristic function of the set E € 2 , then , if { E } is a disjoint sequence of measurable subsets of S and UE , Eo , it follows from III.6.16 that the series converging in the norm of L ...
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