Linear Operators: General theory |
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Page 20
... sequence { a } in a metric space is a Cauchy sequence if lim , 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 . 6 ...
... sequence { a } in a metric space is a Cauchy sequence if lim , 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 . 6 ...
Page 68
... sequence { x } such that { x * x } is a 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 ...
... sequence { x } such that { x * x } is a 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 ...
Page 345
... sequence ( a sequence converging weakly to fin 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 fin 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
B Topological Preliminaries | 10 |
Metric Spaces | 23 |
Product Spaces | 31 |
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 compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations Doklady Akad Duke Math E₁ elements ergodic exists extension f₁ function defined function f Hausdorff space Hence Hilbert space homomorphism inequality 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 o-field 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 topological space u-integrable u-measurable u-null uniformly unit sphere valued function vector space weakly compact zero ΕΕΣ