Linear Operators: General theory |
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Page 20
... sequence { a } is said to be convergent if ana for some a . A sequence { a } in a metric space is a Cauchy sequence if limm , n o ( am , an ) = 0. If every Cauchy se- quence is convergent , a metric space is said to be complete . The ...
... sequence { a } is said to be convergent if ana for some a . A sequence { a } in a metric space is a Cauchy sequence if limm , n o ( am , an ) = 0. If every Cauchy se- quence is convergent , a metric space is said to be complete . The ...
Page 68
... Cauchy sequence of scalars for each x * € X * is called a weak Cauchy sequence . The space X is said to be weakly complete if every weak Cauchy sequence has a weak limit . In Chapter V , a topology is introduced in certain linear spaces ...
... Cauchy sequence of scalars for each x * € X * is called a weak Cauchy sequence . The space X is said to be weakly complete if every weak Cauchy sequence has a weak limit . In Chapter V , a topology is introduced in certain linear spaces ...
Page 122
... sequence of functions in L ( S , Σ , μ , X ) and let f be a function on S to X. Then f is in L , and fn - fp ... Cauchy sequence in L1 ( S ) . For * ɛ > 0 there is , by ( iii ) , a set E , with v ( μ , E ) < ∞ such that 8 \ En −8m \ 1 ...
... sequence of functions in L ( S , Σ , μ , X ) and let f be a function on S to X. Then f is in L , and fn - fp ... Cauchy sequence in L1 ( S ) . For * ɛ > 0 there is , by ( iii ) , a set E , with v ( μ , E ) < ∞ such that 8 \ En −8m \ 1 ...
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 continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional 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 Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ