Linear Operators: General theory |
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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 + I { x + yye I } . If we define the operations ( x + 1 ) + ( y + 1 ) ( x + y ) +1 , ( x + 1 ) ( y + I ) = = xy + 1 , = the ...
... 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 + I { x + yye I } . If we define the operations ( x + 1 ) + ( y + 1 ) ( x + y ) +1 , ( x + 1 ) ( y + I ) = = xy + 1 , = the ...
Page 254
... seen that v is equivalent to vg and thus that vg is in V. Since { v } is a basis , the vector u , has an expansion of the form u ( u , v ) vg , so that u , is in the closed linear manifold de- termined by those vs with ( u , v ) 0 ...
... seen that v is equivalent to vg and thus that vg is in V. Since { v } is a basis , the vector u , has an expansion of the form u ( u , v ) vg , so that u , is in the closed linear manifold de- termined by those vs with ( u , v ) 0 ...
Page 690
... seen that g is in L ,. By Lemma 6.4 there is a positive operator P1 ≤ 1 , P∞ ≤1 , and \ T " ( f , · ) | ≤ P " ( \ ƒ ( · ) | , · ) , n = P in L1 with 1 , 2 , .... By the Riesz convexity theorem we have also P , ≤ 1 , and , since Pr ...
... seen that g is in L ,. By Lemma 6.4 there is a positive operator P1 ≤ 1 , P∞ ≤1 , and \ T " ( f , · ) | ≤ P " ( \ ƒ ( · ) | , · ) , n = P in L1 with 1 , 2 , .... By the Riesz convexity theorem we have also P , ≤ 1 , and , since Pr ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space 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 Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ