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 K1 = K co { xn , xn + ...
... 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 K1 = K co { xn , xn + ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
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 closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism 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 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 Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ