Linear Operators: General theory |
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Page 36
... vector space , linear space , or vector space over a field is an additive group together with an operation m : xX → X , written as m ( x , x ) = xx , which satisfy the following four conditions : ( i ) a ( x + y ) = ax + ay , ( ii ) ...
... vector space , linear space , or vector space over a field is an additive group together with an operation m : xX → X , written as m ( x , x ) = xx , which satisfy the following four conditions : ( i ) a ( x + y ) = ax + ay , ( ii ) ...
Page 49
... linear vector spaces will be over the field of real or the field of complex numbers . A real vector space is one for which the field is the set of real numbers ; a complex vector space is one for which is the set of complex numbers ...
... linear vector spaces will be over the field of real or the field of complex numbers . A real vector space is one for which the field is the set of real numbers ; a complex vector space is one for which is the set of complex numbers ...
Page 58
... linear space is an F - space under each of two metrics , and if one of the corresponding topologies contains the other , the two topologies are equal . - ( X , T1 ) , X2 - PROOF . Let t1 , to be metric topologies in the linear space X ...
... linear space is an F - space under each of two metrics , and if one of the corresponding topologies contains the other , the two topologies are equal . - ( X , T1 ) , X2 - PROOF . Let t1 , to be metric topologies in the linear space X ...
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