Download PDF by Inc. BarCharts: Calculus 2

By Inc. BarCharts

ISBN-10: 1572224754

ISBN-13: 9781572224759

Fundamental & differential calculus for complicated scholars.

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Example text

We introduce a system of semi-norms {µcd on End(m by defining µ(T)=supilawil· By the linear part of T, we mean the transformation L(T) e End(m defined by L(T)(Xi) = ~ ai,(k)x(k). k1+ .. l THEOREM 8 (H. Cartan). Let T be an automorphism of~ such that L(T) is the identity and such that the family {T, T 2, T 3 , • • ·} of powers of T is weakly bounded. Then T is the identity. 32 HOLOMORPHIC FUNCTIONS AND MAPPINGS ON A BOUNDED DOMAIN PROOF (cf. [5: pp. 13-14]). X; mod m2, where m is the maximal ideal of(').

On the other hand, let P[~d. = N- 1· ~ (P, or) j( , r)d. iz)=N- 1- ~ P,(z·r)j(z, r)d TEI' is an automorphic form given by a series (1). - 1• Now f is fixed. ) goes to zero by Cauchy's integral formula applied to a fixed, small polycylindrical neighborhood of 0. t)}t=i .... , k· Thus we have proved THEOREM 10 (Giraud, H. Cartan [15c]). eD belong to distinct orbits of I' and let a positive integer l be given. Adopt notation as above for each point ai, attaching the subscript i to those objects associated to ai.

Let x 0 E X- M. Then there exists a continuous linear functional f on X such that f(x 0) 0, fl M = 0. * Our general reference for the matters in this section is [64]. 28 HOLOMORPHIC FUNCTIONS AND MAPPINGS ON A BOUNDED DOMAIN § 2. Bounded families of holomorphic functions Let D be a domain in C" and let C be the family of all complexvalued continuous functions on D. For each compact subset A of D we define a semi-norm 1J,. (f)=sup lf(x)I. {5) x·=:A. subspace of C. Using Cauchy's integral formula in the same manner as in the proof of Montel•s theorem, one proves that a weakly bounded subfamily of LJ(D) is uniformly equi-continuous on any compact subset -of D.

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Calculus 2 by Inc. BarCharts


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