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 ≈ e X 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 ...
... linear space X is a normed linear space , or a normed space , if to each ≈ e X 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 ...
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. The ... 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. The ... 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 E " . 4 COROLLARY . Every linear operator on a finite dimensional normed linear space is continuous . PROOF . Let { b1 , · .... b } be a Hamel basis for the finite dimension- al normed linear space X 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 , · .... b } be a Hamel basis for the finite dimension- al normed linear space X so that every ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ