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 xe X corresponds a real number a called the norm of x which satisfies the conditions : ( i ) │0 = 0 ; | x | > 0 , x # 0 ; ( ii ) x + y≤ x + y , x , y ex ; ( iii ) ...
... linear space X is a normed linear space , or a normed space , if to each xe X corresponds a real number a called the norm of x which satisfies the conditions : ( i ) │0 = 0 ; | x | > 0 , x # 0 ; ( ii ) x + y≤ x + y , x , y ex ; ( 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 \ , 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 \ , x * S * where S * is the closed unit sphere in the space X * conjugate to X. 16 LEMMA ...
Page 245
... space is equivalent to En . 4 COROLLARY . Every linear operator on a finite dimensional normed linear space is continuous . ... 9 ท PROOF . Let ( b1 , · b1 } be a Hamel basis for the finite dimension- al normed linear space X so that ...
... space is equivalent to En . 4 COROLLARY . Every linear operator on a finite dimensional normed linear space is continuous . ... 9 ท PROOF . Let ( b1 , · b1 } be a Hamel basis for the finite dimension- al normed linear space X so that ...
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 ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f g₁ Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ