New PDF release: Seminaire Pierre Lelong (Analyse) Annee 1975-76

By Pierre Lelong

ISBN-10: 3540082565

ISBN-13: 9783540082569

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Show that I f = n I fn . 5 Prove that for every f ∈ S(I, R), f + and f − also belong to S(I, R), and f+ , f≤ I − I f− . f≤ I I 6 Show that if f : I → E is Riemann integrable, then f ∈ B(I, E), that is, every Riemann integrable function is bounded. 7 Suppose f ∈ B(I, R) and Z = (α0 , . . , αn ) is a partition of I. Define n S(f, I, Z) := sup f (ξ) ; ξ ∈ [αj−1 , αj ] (αj − αj−1 ) j=1 and n S(f, I, Z) := inf f (ξ) ; ξ ∈ [αj−1 , αj ] (αj − αj−1 ) j=1 and call them the upper and lower sum of f over I with respect to the partition Z.

For the sequently, α belongs to L T (I, E), E , and we have α constant function 1 ∈ T (I, R) with value 1, we have claim follows. 1(e)). 2 β that the integral α is a continuous, linear map from T (I, E) to E. 6 to get a unique continuous linear extension of α into the Banach space S(I, E). We denote this extension using the same notation, so that β ∈ L S(I, E), E . 6, it follows that β β f = lim α n fn in E α for f ∈ S(I, E) , 20 VI Integral calculus in one variable where (fn ) is an arbitrary sequence of staircase functions that converges uniformly β to f .

12). (b) Suppose s ∈ R[X]. Every zero z ∈ C of s has a complex conjugate zero z with the same multiplicity. 3, s(z) = s(z). (c) Suppose p, q ∈ R[X] and z ∈ C\R is a zero of q with multiplicity m. 9). Then bk = ak . Proof From (b) we know that z is also a zero of q with multiplicity m. Therefore bk is uniquely determined. For x ∈ C\{z1 , . . 9) that n mj j=1 k=1 p(x) ajk p = = (x) = (x − zj )k q(x) q n mj j=1 k=1 ajk . 8. Suppose now that r := p/q for p, q ∈ R[X] with deg(p) < deg(q) is a real rational function.

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Seminaire Pierre Lelong (Analyse) Annee 1975-76 by Pierre Lelong

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