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 , 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 ...
... present theorem is a corollary of Theorem 6.18 . Q.E.D. 8. The Spaces L „ ( 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 ...
Page 286
... present , that u ( S ) < ∞ . If XE is the characteristic function of the set E € Σ , then , if { E } is a disjoint sequence of measurable subsets of S and U11 E , = Eo , it follows from III.6.16 that 1 ZE , = XE ,, the series Σ1 ...
... present , that u ( S ) < ∞ . If XE is the characteristic function of the set E € Σ , then , if { E } is a disjoint sequence of measurable subsets of S and U11 E , = Eo , it follows from III.6.16 that 1 ZE , = XE ,, the series Σ1 ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ