Linear Operators: General theory |
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Page 26
... neighborhood N1 of y , wx x there exists a neighborhood N of x , such that g ( N ) CN . The follow- ing is a related , but more general notion : Let A be a set , and let X and Y be topological spaces . Let f : AX , and g : A → Y . Then ...
... neighborhood N1 of y , wx x there exists a neighborhood N of x , such that g ( N ) CN . The follow- ing is a related , but more general notion : Let A be a set , and let X and Y be topological spaces . Let f : AX , and g : A → Y . Then ...
Page 51
... neighborhood of f ( x ) . If U is a neighborhood of a such that f ( U ) CV ( y - 1x ) , then Ux - ly is a neighborhood of y such that f ( Ux − 1y ) = f ( U ) f ( x ̄1y ) C Vƒ ( y ̄1x ) f ( x − 1y ) = V. Therefore , f is con- tinuous ...
... neighborhood of f ( x ) . If U is a neighborhood of a such that f ( U ) CV ( y - 1x ) , then Ux - ly is a neighborhood of y such that f ( Ux − 1y ) = f ( U ) f ( x ̄1y ) C Vƒ ( y ̄1x ) f ( x − 1y ) = V. Therefore , f is con- tinuous ...
Page 56
... neighborhood M of 0 such that M - MCG . For every xe X , x / n → 0 , and so xe nM for large n . Thus 00 X = UnM , Y ... neighborhood of 0. Thus the closure of the image of a neighborhood of the origin contains a neighborhood of the ...
... neighborhood M of 0 such that M - MCG . For every xe X , x / n → 0 , and so xe nM for large n . Thus 00 X = UnM , Y ... neighborhood of 0. Thus the closure of the image of a neighborhood of the origin contains a neighborhood of the ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz 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