I am studying the Ito integral from the book Stochastic Analysis for Finance from Geon Ho Choe. However an argument for the Ito integral for non-simple processes is not clear to me.
In his argument he defines $L^2(\Omega) = \left \{ X \ : \ ||X||_2 < \infty \right \}$ with $||X||_2 := \left ( \int_{\Omega}|X|^2d\mathbb{P} \right )^{\frac{1}{2}}$ and claims that this is a norm (I believe this should be a semi-norm) and that the space $L^2(\Omega)$ with this norm is complete.
He then introduces the space of simple processes $ \mathcal{H}_0^2$ and on this space he introduces the norm (I think this should be a semi-norm as well) $||f||_{\mathcal{H}_0^2} = \mathbb{E}[\int_0^\infty |f(t,\omega)|^2dt]$.
Then he defines the Ito map $I:\mathcal{H}_0^2 \rightarrow L^2(\Omega)$ where $I(f)$ denotes the Ito integral of the simple process. He claims that this map is linear and continuous and because (according to him) the space $L^2(\Omega)$ is complete the domain of $I$ can be extended to $\mathcal{H}^2$ which contains $\mathcal{H}_0^2$ as a dense subset.
My question is, Is it true that $L^2(\Omega)$ with $||.||_2$ is actually a semi-normed space and if so, does the argument for the extension of the domain of $I$ stil hold?