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 X / M . It is a linear transformation . Unless otherwise stated , the coefficient field Ø for a linear space X will be either the field of real numbers , in which case X is called a real linear ...
... called the natural homomorphism of X onto X / M . It is a linear transformation . Unless otherwise stated , the coefficient field Ø for a linear space X will be either the field of real numbers , in which case X is called a real linear ...
Contents
B Topological Preliminaries | 10 |
Metric Spaces | 23 |
Product Spaces | 31 |
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 compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations Doklady Akad Duke Math E₁ elements ergodic exists extension f₁ function defined function f Hausdorff space Hence Hilbert space homomorphism inequality 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 o-field 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 topological space u-integrable u-measurable u-null uniformly unit sphere valued function vector space weakly compact zero ΕΕΣ