# Is the space of continuous functions a Cauchy complete?

I am so new to functional analysis so I am looking for an answer of a confusion I am having right now in my mind because I have seen many different answers for the question I am gonna ask below. I hope you will reply as simple as possible because I am not a Mathematician. Thanks for the help in advance...

Is the space of continuous functions $C^{0}$ a Cauchy complete? Therefore is it a Hilbert space or not?

There's a thesis online, which says that this space is not Cauchy complete and is therefore not a Hilbert space. $L^2$ square integrable functions space is the Cauchy completion of the function space $C^0$ and in other words, contnuous functions on domain $X$ are dense in $L^{2}(X)$.

However, I run into some documents which support that the space of continuous functions is a Cauchy complete.

• "Some documents" may be talking about a metric other than "uniform convergence" ... Or they may be talking about continuous functions defined on some space other than a compact interval. Who knows? – GEdgar Mar 24 '14 at 13:57