Linear Operators: General theory |
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Page 20
... 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 three lemmas are immediate consequences of the definitions ...
... 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 three lemmas are immediate consequences of the definitions ...
Page 68
... 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 such a ...
... 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 such a ...
Page 345
... sequence ( a sequence converging weakly to ƒ in A ( D ) ) if and only if it is uniformly bounded and conver- ges at each point of the boundary of D ( to f ) . Show that a subset of A ( D ) is conditionally compact if and only if it is ...
... sequence ( a sequence converging weakly to ƒ in A ( D ) ) if and only if it is uniformly bounded and conver- ges at each point of the boundary of D ( to f ) . Show that a subset of A ( D ) is conditionally compact if and only if it is ...
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