Linear Operators: General theory |
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... complex number systems . By the extended real number system we mean the real numbers with the symbols + ∞ and ∞ adjoined ; by the extended complex number system we mean the complex numbers with the single symbol ∞o adjoined . If A is ...
... complex number systems . By the extended real number system we mean the real numbers with the symbols + ∞ and ∞ adjoined ; by the extended complex number system we mean the complex numbers with the single symbol ∞o adjoined . If A is ...
Page 238
... complex valued functions f , g defined on a specified domain S. Here , addition and multiplication are understood to ... numbers when the real and the complex B - spaces under consideration actually require separate treatment . The proof ...
... complex valued functions f , g defined on a specified domain S. Here , addition and multiplication are understood to ... numbers when the real and the complex B - spaces under consideration actually require separate treatment . The proof ...
Page 556
... complex numbers 2 such that I - T is not one - to - one . The index v ( 2 ) of a complex number 2 is the smallest non - negative integer v such that ( 21 - T ) x = 0 for every vector a for which ( 1 - T ) + 1x = 0 . It follows that if 2 ...
... complex numbers 2 such that I - T is not one - to - one . The index v ( 2 ) of a complex number 2 is the smallest non - negative integer v such that ( 21 - T ) x = 0 for every vector a for which ( 1 - T ) + 1x = 0 . It follows that if 2 ...
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