Linear Operators: General theory |
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Page 59
... linear space X is a normed linear space , or a normed space , if to each a ex corresponds a real number | x | called the norm of a which satisfies the conditions : ( i ) 0 = 0 ; | x | > 0 , x 0 ; ( ii ) x + y≤ x + y , x , y € X ; ( iii ) ...
... linear space X is a normed linear space , or a normed space , if to each a ex corresponds a real number | x | called the norm of a which satisfies the conditions : ( i ) 0 = 0 ; | x | > 0 , x 0 ; ( ii ) x + y≤ x + y , x , y € X ; ( iii ) ...
Page 65
... normed linear space X , there is an x * X * with │x * | = 1 and x * x = = \ x \ . PROOF . Apply Lemma 12 with Y 0 ... linear space X , = sup x * x . ** S * where S * is the closed unit sphere in the space X * conjugate to X. 16 LEMMA ...
... normed linear space X , there is an x * X * with │x * | = 1 and x * x = = \ x \ . PROOF . Apply Lemma 12 with Y 0 ... linear space X , = sup x * x . ** S * where S * is the closed unit sphere in the space X * conjugate to X. 16 LEMMA ...
Page 245
... space is equivalent to E " . 4 COROLLARY . Every linear operator on a finite dimensional normed linear space is continuous . PROOF . Let { b1 , . . . , bn } be a Hamel basis for the finite dimension- al normed linear space so that every ...
... space is equivalent to E " . 4 COROLLARY . Every linear operator on a finite dimensional normed linear space is continuous . PROOF . Let { b1 , . . . , bn } be a Hamel basis for the finite dimension- al normed linear space so that every ...
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