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I've noticed that it's often interesting to consider what types of properties of a topology are preserved or lost when mapping between different types of spaces. This led me to wonder about spaces of functions themselves.

In this case, suppose I have two metric spaces, $(S,c)$ compact and $(T,d)$ complete, and I denote by $C(S,T)$ the set of all continuous functions from $S\to T$. I can put a metric on $C(S,T)$ defined by $\rho(f,g)=\sup_{s\in S}d(f(s),g(s))$. With this in place, is it true that $(C(S,T),\rho)$ is also complete?

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It's a standard result that it is. What have you tried so far? –  Olivier Bégassat Oct 4 '11 at 0:35

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up vote 3 down vote accepted

Under these assumptions it is true that $(C(S,T),\rho)$ is complete.

Outline of the proof:

  • Pick a Cauchy sequence $(f_n)$ in $C(S,T)$.
  • Use the Cauchyness and the completeness of $T$ to show that the sequence $(f_n(s))$ converges for each $s\in S$.
  • Show that the function $f$ defined by $f(s)=\lim_{n\to\infty}f_n(s)$ is continuous and $(f_n)$ converges to $f$.

Munkres' Topology (2nd edition) has a complete proof (Theorem 43.6., page 267).

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And show that $f$ is itself continuous. –  Olivier Bégassat Oct 4 '11 at 1:01
    
@Olivier Bégassat: Thanks! –  LostInMath Oct 4 '11 at 1:07
    
Thanks for the proof and reference. –  Gotye Oct 5 '11 at 19:36

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