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.
This question has been addressed ad infinitum.
For proof that $C^0$ is complete with the supremum norm (hence is a Banach space) see here: How to show that $C=C[0,1]$ is a Banach space
For proof that the supremum norm arises from no inner product (and hence is not a Hilbert space) see here: $C[0,1]$ is not Hilbert space