Linear Operators: General theory |
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Page 36
... vector space , linear space , or vector space over a field Þ is an additive group together with an operation m : × X → 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 Þ is an additive group together with an operation m : × X → X , written as m ( x , x ) = xx , which satisfy the following four conditions : ( i ) a ( x + y ) = ax + ay , ( ii ) ...
Page 37
... vectors in the domain of T. Thus a linear transformation on a linear space X is a homomorphism on the additive group X which commutes with the operations of multiplication by scalars . If T : X → Y and U : Y → 3 are linear ...
... vectors in the domain of T. Thus a linear transformation on a linear space X is a homomorphism on the additive group X which commutes with the operations of multiplication by scalars . If T : X → Y and U : Y → 3 are linear ...
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 ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional 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 Russian 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