Linear Operators: General theory |
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Page 34
... called multiplication . The element ab is called the product of a and b . The product ab is required to satisfy the following conditions : ( i ) a ( bc ) = ( ab ) c , a , b , c e G ; ( ii ) there is an element e in G , called the ...
... called multiplication . The element ab is called the product of a and b . The product ab is required to satisfy the following conditions : ( i ) a ( bc ) = ( ab ) c , a , b , c e G ; ( ii ) there is an element e in G , called the ...
Page 35
... called a homo- morphism if h ( ab ) = h ( a ) h ( b ) . A one - to - one homomorphism is called an isomorphism . If h : A → B is an isomorphism and if h ( A ) = B , then A and B are said to be isomorphic , or A is said to be isomorphic ...
... called a homo- morphism if h ( ab ) = h ( a ) h ( b ) . A one - to - one homomorphism is called an isomorphism . If h : A → B is an isomorphism and if h ( A ) = B , then A and B are said to be isomorphic , or A is said to be isomorphic ...
Page 38
... called the natural homomorphism of X onto XM . It is a linear transformation . Unless otherwise stated , the coefficient field for a linear space X will be either the field of real numbers , in which case X is called a real linear space ...
... called the natural homomorphism of X onto XM . It is a linear transformation . Unless otherwise stated , the coefficient field for a linear space X will be either the field of real numbers , in which case X is called a real linear space ...
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