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 a which satisfies the conditions : € ( i ) │0 = 0 ; | 0 | 0 ; x > 0 , x 0 ; = ( ii ) │x + y | ≤ | x | + | y ...
... 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 a which satisfies the conditions : € ( i ) │0 = 0 ; | 0 | 0 ; x > 0 , x 0 ; = ( ii ) │x + y | ≤ | x | + | y ...
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 , | x | = sup x * x , x * € S * where S * is the closed unit sphere in the space X * conjugate to X. 16 ...
... 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 , | x | = sup x * x , x * € S * where S * is the closed unit sphere in the space X * conjugate to X. 16 ...
Page 245
... space is equivalent to E " . COROLLARY . Every linear operator on a finite dimensional normed linear space is continuous . ... 9 PROOF . Let { b1 , b } be a Hamel basis for the finite dimension- al normed linear space X so that every x ...
... space is equivalent to E " . COROLLARY . Every linear operator on a finite dimensional normed linear space is continuous . ... 9 PROOF . Let { b1 , b } be a Hamel basis for the finite dimension- al normed linear space X so that every x ...
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 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 disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ