Linear Operators: General theory |
From inside the book
Results 1-3 of 43
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 ( am , an ) = 0. If every Cauchy se- quence is convergent , a metric space is said to be complete . The next ...
... 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 ( am , an ) = 0. If every Cauchy se- quence is convergent , a metric space is said to be complete . The next ...
Page 68
... Cauchy sequence of scalars for each * 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 in ...
... Cauchy sequence of scalars for each * 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 in ...
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 ε > 0 there is , by ( iii ) , a set E , with v ( u , E ) < ∞ such that \ 8n — 6m'1 ≤ 2 ...
... 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 ε > 0 there is , by ( iii ) , a set E , with v ( u , E ) < ∞ such that \ 8n — 6m'1 ≤ 2 ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
59 other sections not shown
Other editions - View all
Common terms and phrases
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