Linear Operators: General theory |
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Page 36
... vector space , linear space , or vector space over a field Ø is an additive group together with an operation m : Þ × X written as m ( x , x ) = ax , which satisfy the following four conditions : → X , ( i ) x ( x + y ) = ax + xy , αεΦ ...
... vector space , linear space , or vector space over a field Ø is an additive group together with an operation m : Þ × X written as m ( x , x ) = ax , which satisfy the following four conditions : → X , ( i ) x ( x + y ) = ax + xy , αεΦ ...
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 TX and U : Y - > Y 3 are linear transformations ...
... 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 TX and U : Y - > Y 3 are linear transformations ...
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
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space 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 Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ