The following is not completely clear to me: Let $\mathcal{H}^2$ be the set of all RCLL martingales $ M$ which are bounded in $L^2$, i.e. $\sup_tE[M_t^2]<\infty$. Then it is a standard result that $\mathcal{H}^2$ becomes a Hilbert space with the norm $\|M_\infty\|_{L^2}$. One can show that the subspace of all continuous martingales in $\mathcal{H}^2$, denoted with $\mathcal{H}_c^2$, is a closed subspace. Then the space of $\mathcal{H}^2_d$ is called the space of all purely discontinuous martingales and it is defined to be the orthogonal complement of $\mathcal{H}^2_c$, i.e. all $M\in\mathcal{H}^2$ such that
$$E[M_\infty N_\infty]=0$$ for all $N\in\mathcal{H}^2_c$. Of course you can denote with $\mathcal{H}^2_{loc}$ the localized version of the above spaces.
Now a general local martingale $M$ is called purely discontinuous if $MN$ is a local martingale for very continuous local martingale $N$. I was wondering what is the connection between these definitions. I assume that all local martingales satisfy $M_0=0$. Why are there two different definitions for the same "name"?