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 Q ( 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 Q ( 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 a * 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 a * 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 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