Suppose that H is a nonempty compact subset of X and that Y is a Euclidean space.
a) If f: H †’ y is continuous, prove that

is finite and there exists an x0 ˆˆ H such that ||f(x0))||γ = ||f||H.
b) A sequence of functions fk: H †’ Y is said to converge uniformly on H to a function f: H †’ Y if and only if given ε > 0 there is an N ˆˆ N such that
k > N and x ˆˆ H imply ||fk(x) - f(x)||γ Show that ||fk - f||H †’ 0 as A: †’ ˆž if and only if fk †’ f uniformly on H as k †’ 00.
c) Prove that a sequence of functions fk converges uniformly on H if and only if, given ε > 0, there is an N ˆˆ N such that
k, j > N implies ||fk - fj||H

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