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 ≤oy in the ordering of that E 。. It is clear that if ( UE , ≤ ' ) belongs to & it is an upper ...
... 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 ≤oy in the ordering of that E 。. It is clear that if ( UE , ≤ ' ) belongs to & it is an upper ...
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 v is equivalent to vg and thus that ϋβ is in V. Since { v } is a basis , the vector u has an expansion of the ( u , v ) , so that u , is in the closed linear manifold de- α form u = 0. Since such are in V we have Similarly sp ...
... seen that v is equivalent to vg and thus that ϋβ is in V. Since { v } is a basis , the vector u has an expansion of the ( u , v ) , so that u , is in the closed linear manifold de- α form u = 0. Since such are in V we have Similarly sp ...
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 B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ