Is the space of $k$-forms on a compact Riemannian manifold $M$ with the inner product given by $$(\alpha,\beta)=\int \alpha \wedge *\beta=\int g(\alpha,\beta)dv$$ which is called «$L^2$ product in $\Omega^k$» equivalent to the space $L^2$ on $M$?
Sorry I didn't understand properly what was the space $L^2(\Omega^p(M))$ so my question was misleading. Now I think I do, and it is the space of $p$-forms that have coefficients in $L^2$ in a any chart. My question would be then that if we extend what I called the «L^2 product» to the direct sum $\Omega(M):=\sum \Omega^p$ by saying that $\Omega^p$ is orthogonal to $\Omega^q$ for $p \neq q$ and then to $L^2(\Omega(M))$ (I think you can do this because on a compact manifold the smooth forms are dense in the $L^2$ ones right?) then what it is the relation of this space with the space $L^2(M)$. Is it a direct sum of copies of it?? thanks