This seems to be a very easy excercise, but I am completely stuck:(adsbygoogle = window.adsbygoogle || []).push({});

Prove that in C([0,1]) with the metric

[tex] \rho(f,g) = (\int_0^1|f(x)-g(x)|^2 dx)^{1/2} [/tex]

a subset

[tex]A = \{f \in C([0,1]); \int_0^1 f(x) dx = 0\}[/tex] is closed.

I tried to show that the complement of A is open - it could be easily done if the metric was [tex] \rho(f,g) = sup_{x \in [0,1]}|f(x)-g(x)| [/tex] - but with the integral metric it's not that easy.

Am I missing something?

Thanks for any help.

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# Closed subset of a metric space

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