Prove the second requirement (ii) is necessary for the statement to be true. Theorem 14. A family F of continuous functions with values i a metricspace S is normal in the region 2 of the complex plane if and only if(i) F is equicontinuous on every compact set E C;(ii) for any z eN the values f(2), ƒ e F, lie in a compact subset of S.

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Theorem 14. A family F of continuous functrons Wiu. space S is normal in the region 2 of the complex plane if and only if (i) F is equicontinuous on every compact set EC; (ii) for any z eN the values f(2), fe F, lie in a compact subset of S. es in a metrie

Theorem 14. A family F of continuous functrons Wiu. space S is normal in the region 2 of the complex plane if and only if (i) F is equicontinuous on every compact set EC; (ii) for any z eN the values f(2), fe F, lie in a compact subset of S. es in a metrie

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.2: Integral Domains And Fields

Problem 16E: Prove that if a subring R of an integral domain D contains the unity element of D, then R is an...

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Question

Prove the second requirement (ii) is necessary for the statement to be true.

Theorem 14. A family F of continuous functions with values i a metric
space S is normal in the region 2 of the complex plane if and only if
(i) F is equicontinuous on every compact set E C;
(ii) for any z eN the values f(2), ƒ e F, lie in a compact subset of S.

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