Linear Operators: General theory |
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Page 8
... subsets of a set has the property that A € E if and only if every finite subset of A belongs to & , then & contains a maximal element . 3 Derive Theorem 6 from the statement of Exercise 2 . 4 Show that any one of the following implies ...
... subsets of a set has the property that A € E if and only if every finite subset of A belongs to & , then & contains a maximal element . 3 Derive Theorem 6 from the statement of Exercise 2 . 4 Show that any one of the following implies ...
Page 17
... subset of a normal space has a real continuous extension defined on the whole space . PROOF . The only case where the theorem does not apply imme- diately is the case where ƒ is unbounded . If ƒ is real and continuous on the closed set ...
... subset of a normal space has a real continuous extension defined on the whole space . PROOF . The only case where the theorem does not apply imme- diately is the case where ƒ is unbounded . If ƒ is real and continuous on the closed set ...
Page 440
... subset of K which furnishes a lower bound for A1 . It follows by Zorn's lemma that contains a minimal element A. Suppose that 4 contains two distinct points p and q . Then , by 2.13 , there is a functional a * € X * such that Rx * ( p ) ...
... subset of K which furnishes a lower bound for A1 . It follows by Zorn's lemma that contains a minimal element A. Suppose that 4 contains two distinct points p and q . Then , by 2.13 , there is a functional a * € X * such that Rx * ( p ) ...
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