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Continuous Functions

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Theorem: Let E and E' be metric spaces, with E compact and E' complete. Then the set of all continuous functions from E to E', with the distance between two such functions f and g taken to be max {d'( f(p), g(p) ) : p is an element of E} is a complete metric space. A sequence of points of this metric space converges if and only if it is a uniformly convergent sequence of functions on E.

Prove the analog of this theorem when E is not compact but with a restriction to bounded continuous functions, the distance between two such functions f and g being taken as l.u.b. {d'( f(p), g(p) ) : p is an element of E}.

Do the same thing for bounded functions from E to E' that are not necessarily continuous.

What is the relation between the two metric spaces so obtained?

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This solution helps explore continuous functions within the context of real analysis.

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