Linear Operators: General theory |
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Page 88
... condition is sufficient . Bonsall [ 1 ] showed that the separa- bility condition cannot be dropped . Ingleton [ 1 ] has given conditions for the Hahn - Banach theorem to hold when the field of scalars is non - Archimedean . ( See also ...
... condition is sufficient . Bonsall [ 1 ] showed that the separa- bility condition cannot be dropped . Ingleton [ 1 ] has given conditions for the Hahn - Banach theorem to hold when the field of scalars is non - Archimedean . ( See also ...
Page 131
... condition is obvious . To prove the sufficiency of the condition we observe first that a set function 2 satisfies this condition if and only if the positive and negative varia- tions of its real and imaginary parts satisfy the same ...
... condition is obvious . To prove the sufficiency of the condition we observe first that a set function 2 satisfies this condition if and only if the positive and negative varia- tions of its real and imaginary parts satisfy the same ...
Page 487
... condition is satisfied . Q.E.D. 6. Operators with Closed Range It was observed in Lemma 2.8 that the closure of the ... condition , and define a ( possibly dis- continuous ) linear functional y on Yo UX by the formula y * ( Ux ) = x ...
... condition is satisfied . Q.E.D. 6. Operators with Closed Range It was observed in Lemma 2.8 that the closure of the ... condition , and define a ( possibly dis- continuous ) linear functional y on Yo UX by the formula y * ( Ux ) = x ...
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