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What is the completion of the metric space of continuous functions with metric $$d(f,g)=\sqrt{\int\limits_a^b(f-g)^2}$$

$f,g$ are function from $[a,b]$ to $\mathbb{R}$

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  • $\begingroup$ not every continuous function on $\mathbb R$ is integrable, let alone square integrable. For example any non zero polynomial. $\endgroup$ – user251257 Sep 14 '15 at 20:31
  • $\begingroup$ Sorry, the domain is the compact set $[a,b]$ $\endgroup$ – José Sep 14 '15 at 20:31
  • $\begingroup$ then $L^2[a,b]$ $\endgroup$ – user251257 Sep 14 '15 at 20:32
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Note that right now problem is not well posed. You really need to specify the domain of definition of those functions. If it is not compact (for example whole real line) the integral will diverge in general. So your metric wouldn't be well defined. However in majority of relevant cases it would be space of square integrable functions in Lebesgue sense, called $L^2$. That's actually one of the reasons why Lebesgue integral is better than Riemann's.

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  • $\begingroup$ $f,g$ are function from $\mathbb{R}$ to $\mathbb{R}$ $\endgroup$ – José Sep 14 '15 at 20:28
  • $\begingroup$ Then clearly if you take $f=1$ and $g=2$ (constant functions) their distance will be infinite. So this is not really a metric space. $\endgroup$ – Blazej Sep 14 '15 at 20:31
  • $\begingroup$ Sorry, the domain is the compact set $[a,b]$ $\endgroup$ – José Sep 14 '15 at 20:32
  • $\begingroup$ Then it would be $L^2(a,b)$ that is space of all functions on this interval that are regular enough to be integrated and the integral of their square is finite. In particular it contains plenty of discontinuous functions. $\endgroup$ – Blazej Sep 14 '15 at 20:35

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