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Is $F=\left\lbrace f: f\in C_{\left[0, 1\right]}, f\left(0\right)=f\left(1\right)\right\rbrace$ complete metric space?

I know how to prove that $C_{\left[0, 1\right]}$ is complete. But I don't know how to use condition that $f\left(0\right)=f\left(1\right)$.

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$F$ is a closed subspace of the complete space $\mathcal C_{\left[0, 1\right]}$. Hence it is complete.

Why is $F$ closed? Because $F$ is the inverse image of (the closed subset) $\{0\}$ under the continuous map $f \mapsto f(1)-f(0)$.

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    $\begingroup$ Of course, you are assuming here that your map is continuous under the as-yet-unspecified metric on $\mathcal C_{[0,1]}$. But that is a very safe assumption. $\endgroup$ – Paul Sinclair Jun 28 '18 at 16:38
  • $\begingroup$ I imagine yes... This is a very usual norm on that space! $\endgroup$ – mathcounterexamples.net Jun 28 '18 at 16:40
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You only need to show that $F$ is a closed subspace of the complete space $C[0,1]$.

Consider a sequence $(f_n)_{n\in\mathbb{N}} \subset F$ with $f_n \stackrel{n\rightarrow \infty}{\Longrightarrow} f$ in $C[0,1]$.

Then $f_n(0) \stackrel{n\rightarrow \infty}{\longrightarrow} f(0)$ and $f_n(1) \stackrel{n\rightarrow \infty}{\longrightarrow} f(1)$.

So, $f \in F$ as well, because $0 = f_n(0)-f_n(1) \stackrel{n\rightarrow \infty}{\longrightarrow} f(0)-f(1) = 0$.

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$F$ is closed in the pointwise topology on $C([0,1])$ and hence also in the sup-metric topology. A closed subspace of a complete metric space is complete in the inherited metric.

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You have to take a Cauchy sequence in $F$ and show that it converges to a limit in $F$. Since $C([0,1])$ is complete, then taking such a sequence converges to a function $f$ in $C([0,1])$.

Now using the condition for the endpoints for each entry of the sequence you get that $f(0)=f(1)$, hence is in $F$. Thus your subspace is complete.

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  • First try to prove that closed subspace of a metric space is complete.

  • Then try to show it is a closed subspace.

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  • $\begingroup$ I think it should better be labeled as a hint. $\endgroup$ – user 170039 May 21 at 8:13

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