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 433
... condition implies that K is bounded . For otherwise , there exists an a * € X * such that x * ( K ) is an unbounded ... condition . Further , the condition implies that K is closed , for if xn → xq , xn € K , we set Kn K ○ co { xn xn + ...
... condition implies that K is bounded . For otherwise , there exists an a * € X * such that x * ( K ) is an unbounded ... condition . Further , the condition implies that K is closed , for if xn → xq , xn € K , we set Kn K ○ co { xn xn + ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ