Linear Operators: General theory |
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Page 287
... evident from the Hölder inequality that any ge L , deter- mines an x * L * satisfying ( i ) , so that the mapping x * ↔ g is a one - to - one isometric map of L * onto L. Since the linearity of the map is evident , the theorem is ...
... evident from the Hölder inequality that any ge L , deter- mines an x * L * satisfying ( i ) , so that the mapping x * ↔ g is a one - to - one isometric map of L * onto L. Since the linearity of the map is evident , the theorem is ...
Page 299
... evident that • + 00 lim ( + \ x ( x + y ) —x ( y ) | Pdy = 0 81 if % is the characteristic function of a finite interval . Thus lim S ± \ g ; ( x + y ) — g ; ( y ) \ " dy = 0 for each function g1 ; and hence 04x ∞ + . • + ∞ lim sup ...
... evident that • + 00 lim ( + \ x ( x + y ) —x ( y ) | Pdy = 0 81 if % is the characteristic function of a finite interval . Thus lim S ± \ g ; ( x + y ) — g ; ( y ) \ " dy = 0 for each function g1 ; and hence 04x ∞ + . • + ∞ lim sup ...
Page 337
... evident that BV ( I ) = BV ( I ) N. Thus BV ( I ) is isometrically isomorphic to the direct sum of ba ( I , Σ ) and a one - dimensional space . From this , the following theorem is evident ( cf. 9.9 ) . 1 THEOREM . The space BV ( I ) is ...
... evident that BV ( I ) = BV ( I ) N. Thus BV ( I ) is isometrically isomorphic to the direct sum of ba ( I , Σ ) and a one - dimensional space . From this , the following theorem is evident ( cf. 9.9 ) . 1 THEOREM . The space BV ( I ) is ...
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