Linear Operators: General theory |
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Page 262
... v ( μ , S ) < ∞ , f is u - integrable . Since the integral Ssf ( s ) u ( ds ) ... fi Then x * ( fo ) -μ || ≤2ɛ , so that sup | x * ( ƒ ) | = | μ \ . = We ... v ( u , E - F ) < ɛ / 2 , and an open set GE such that v ( μ , G – F ) < ɛ / 4 ...
... v ( μ , S ) < ∞ , f is u - integrable . Since the integral Ssf ( s ) u ( ds ) ... fi Then x * ( fo ) -μ || ≤2ɛ , so that sup | x * ( ƒ ) | = | μ \ . = We ... v ( u , E - F ) < ɛ / 2 , and an open set GE such that v ( μ , G – F ) < ɛ / 4 ...
Page 273
... v ... vts , " 8 Thus ft ( u ) > F ( u ) —ε for u € S. Since f11 , t ( t ) = F ( t ) , we have fi ( t ) = F ( t ) , and hence there is a neighborhood V , of t with ft ( u ) < F ( u ) + ɛ , u € Vt . Let V11 , ... , Vt cover S , and define ...
... v ... vts , " 8 Thus ft ( u ) > F ( u ) —ε for u € S. Since f11 , t ( t ) = F ( t ) , we have fi ( t ) = F ( t ) , and hence there is a neighborhood V , of t with ft ( u ) < F ( u ) + ɛ , u € Vt . Let V11 , ... , Vt cover S , and define ...
Page 302
... fi . This clearly is a partial ordering ( I.2.1 ) in L ,. We now establish completeness ( I.12 ) with respect to ... v sup { 2 } , i.e. , v is the least upper bound of the set { 2 } . Hence v e ca ( S , Σ ) and 0 ≤ λ , ≤ v ≤20 . Since ...
... fi . This clearly is a partial ordering ( I.2.1 ) in L ,. We now establish completeness ( I.12 ) with respect to ... v sup { 2 } , i.e. , v is the least upper bound of the set { 2 } . Hence v e ca ( S , Σ ) and 0 ≤ λ , ≤ v ≤20 . Since ...
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