Linear Operators: General theory |
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Page 169
... zero in u - measure . 5 Show that ( i ) , ( ii ) , and ( iii ) of Theorem 3.6 imply that ƒ is in L , ( S , Z , μ ) and that f - f , converges to zero even if { f } is a general- ized sequence . 6 Let u be bounded . Suppose that the ...
... zero in u - measure . 5 Show that ( i ) , ( ii ) , and ( iii ) of Theorem 3.6 imply that ƒ is in L , ( S , Z , μ ) and that f - f , converges to zero even if { f } is a general- ized sequence . 6 Let u be bounded . Suppose that the ...
Page 204
... zero and since 0 ≤ fn ( $ a , ) ≤ 1 it follows from the dominated convergence theorem ( 6.16 ) that there is a point in S , for which fn ( s ) is defined for all n and for which the se- quence { f ( s ) } does not converge to zero ...
... zero and since 0 ≤ fn ( $ a , ) ≤ 1 it follows from the dominated convergence theorem ( 6.16 ) that there is a point in S , for which fn ( s ) is defined for all n and for which the se- quence { f ( s ) } does not converge to zero ...
Page 452
... zero continuous linear functional tangent to A at p if and only if the cone B with vertex p generated by A is not ... zero continuous linear functional tangent to В at p is a tangent to A at p . Conversely , if ƒ is a non - zero ...
... zero continuous linear functional tangent to A at p if and only if the cone B with vertex p generated by A is not ... zero continuous linear functional tangent to В at p is a tangent to A at p . Conversely , if ƒ is a non - zero ...
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