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 satisfies this condition if and only if the positive and negative varia- tions of its real and imaginary parts satisfy the same condition ...
... condition is obvious . To prove the sufficiency of the condition we observe first that a set function satisfies this condition if and only if the positive and negative varia- tions of its real and imaginary parts satisfy the same condition ...
Page 257
... condition = 0 on an element a in § is equivalent to the condition ( sp ( B ) , x ) = 0. Thus B sp ( B ) and ( iii ) follows by placing M ( B , x ) = = sp ( B ) in ( ii ) . Q.E.D. 5. The Spaces B ( S , E ) and B ( S ) In this section it ...
... condition = 0 on an element a in § is equivalent to the condition ( sp ( B ) , x ) = 0. Thus B sp ( B ) and ( iii ) follows by placing M ( B , x ) = = sp ( B ) in ( ii ) . Q.E.D. 5. The Spaces B ( S , E ) and B ( S ) In this section it ...
Contents
Preliminary Concepts | 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 ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element ergodic exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ