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 x * € X * such that a * ( K ) is an unbounded ... condition . Further , the condition implies that K is closed , for if x , xq , xn € K , we set Kn K co { xn , xn + 1 ...
... condition implies that K is bounded . For otherwise , there exists an x * € X * such that a * ( K ) is an unbounded ... condition . Further , the condition implies that K is closed , for if x , xq , xn € K , we set Kn K co { xn , xn + 1 ...
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