Linear Operators: General theory |
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Page 7
... seen that this upper bound may be defined as ( UE 。, ≤ ' ) , where x ≤'y , whenever x and y both belong to some subset E。€ 80 , and x ≤ y in the ordering ≤ of that E 。. It is clear that if ( UE 。, ≤ ' ) belongs to & it is an ...
... seen that this upper bound may be defined as ( UE 。, ≤ ' ) , where x ≤'y , whenever x and y both belong to some subset E。€ 80 , and x ≤ y in the ordering ≤ of that E 。. It is clear that if ( UE 。, ≤ ' ) belongs to & it is an ...
Page 39
... seen to be a maximal right ideal containing Io . If I is a two - sided ideal in R , let x + I be defined as x + 1 { x + y \ y € I } . If we define the operations ( x + 1 ) + ( y + 1 ) = ( x + y ) +1 , ( x + 1 ) ( y + I ) = xy + 1 ...
... seen to be a maximal right ideal containing Io . If I is a two - sided ideal in R , let x + I be defined as x + 1 { x + y \ y € I } . If we define the operations ( x + 1 ) + ( y + 1 ) = ( x + y ) +1 , ( x + 1 ) ( y + I ) = xy + 1 ...
Page 254
... seen that vg is equivalent to vg and thus that υβ v is in V. Since { v } is a basis , the vector u , has an expansion of the form u ( u , v ) Vg , ( u , v ) , so that u is in the closed linear manifold de- termined by those ϊβ with ( u ...
... seen that vg is equivalent to vg and thus that υβ v is in V. Since { v } is a basis , the vector u , has an expansion of the form u ( u , v ) Vg , ( u , v ) , so that u is in the closed linear manifold de- termined by those ϊβ with ( u ...
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 ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ