I am confusing some definitions. Suppose we have a Cauchy sequence $(f_n) \subset L^2(\Omega,C^0([0,1],\mathbb{R}))$, where $\Omega$ is a measurable space with measure $\mu$ and $C^0([0,1],\mathbb{R})$ is equipped with the uniform norm (that is a Banach space..). What can we conclude and how ?
1) $(f_n)\,\,$converges to a function $f \in L^2(\Omega,C^0([0,1],\mathbb{R}))$ because it is a complete space ? (But I do not know why is it.. I know for $L^p([a,b], \mathbb{R})$).
2) For each $\omega \in \Omega$, $f(\omega)$ is a continuous function on $[0,1]$, because $C^0([0,1],\mathbb{R})$ is a complete space (with the uniform norm) ?
If these two conclusions are true, could someone write explicitly what we need to prove to get these two conclusions ?
I think it is $\int_{\Omega} \sup\limits_{t\in[0,1]} \{(f_n(\omega)(t) - f_m(\omega)(t))^2\} d\mu(\omega) \stackrel{n,m\rightarrow\infty}{\longrightarrow}0$, but I am not sure and even if it is true, I do not understand why.. Could someone help me to clarify these few points, I am really confused for a few days ? Thank you so much. Marcus