Linear Operators: General theory |
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Page 58
... linear , and continuous ( I.8.3 ) . Hence , by Theorem 2 , its inverse pr1 is continuous . Thus T = propr1 is continuous ( I.4.17 ) . Q.E.D. 5 THEOREM . If a linear space is an F - space under each of two metrics , and if one of the ...
... linear , and continuous ( I.8.3 ) . Hence , by Theorem 2 , its inverse pr1 is continuous . Thus T = propr1 is continuous ( I.4.17 ) . Q.E.D. 5 THEOREM . If a linear space is an F - space under each of two metrics , and if one of the ...
Page 417
... linear functional on a linear topological space separates two sets , one of which has an interior point , then the functional is continuous . PROOF . Let X be a linear topological space , and A1 , 42 C X. Let h be a linear functional ...
... linear functional on a linear topological space separates two sets , one of which has an interior point , then the functional is continuous . PROOF . Let X be a linear topological space , and A1 , 42 C X. Let h be a linear functional ...
Page 454
... continuous , TN : C → K is continuous and by the preceding lemma has a fixed point . This fixed point is in K ; it is therefore a fixed point of the mapping T. Q.E.D. 4 LEMMA . Let K be a compact convex subset of a locally convex linear ...
... continuous , TN : C → K is continuous and by the preceding lemma has a fixed point . This fixed point is in K ; it is therefore a fixed point of the mapping T. Q.E.D. 4 LEMMA . Let K be a compact convex subset of a locally convex linear ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space 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 exists f₁ finite dimensional function defined function f g₁ 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-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ