Linear Operators: General theory |
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Page 3
... real function defined by the equations X ( s ) = 1 , se E , and ZE ( s ) = 0 , s ¢ E. XE Sometimes , when the range of a transformation f : A → B is to be emphasized at the expense of ... real numbers , I.1 3 NOTATION AND ELEMENTARY NOTIONS.
... real function defined by the equations X ( s ) = 1 , se E , and ZE ( s ) = 0 , s ¢ E. XE Sometimes , when the range of a transformation f : A → B is to be emphasized at the expense of ... real numbers , I.1 3 NOTATION AND ELEMENTARY NOTIONS.
Page 4
... real numbers , then the symbol lim sup A denotes the infimum of all numbers b with the property that only a finite set of numbers in A exceed b ; the de- finition of the symbol lim inf A is similar . In particular , if A is a sequence ...
... real numbers , then the symbol lim sup A denotes the infimum of all numbers b with the property that only a finite set of numbers in A exceed b ; the de- finition of the symbol lim inf A is similar . In particular , if A is a sequence ...
Page 11
... real numbers . Another base for this topology is obtained by taking a and b to be rational numbers . A subbase for this topology is given by all infinite intervals ( -∞ , a ) , ( b , + ∞ ) , where a and b are either real or rational ...
... real numbers . Another base for this topology is obtained by taking a and b to be rational numbers . A subbase for this topology is given by all infinite intervals ( -∞ , a ) , ( b , + ∞ ) , where a and b are either real or rational ...
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