Linear Operators: General theory |
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Page 36
... linear vector space , linear space , or vector space over a field Þ is an additive group together with an operation m : Ø × X → X , written as m ( x , x ) = ax , which satisfy the following four conditions : α € D , x , y ε X ; = ( i ) ...
... linear vector space , linear space , or vector space over a field Þ is an additive group together with an operation m : Ø × X → X , written as m ( x , x ) = ax , which satisfy the following four conditions : α € D , x , y ε X ; = ( i ) ...
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 . PROOF . Let T1 , T2 be metric topologies in the linear space X for which the spaces ...
... 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 . PROOF . Let T1 , T2 be metric topologies in the linear space X for which the spaces ...
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 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 disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ