I'm currently studying the theory of SPDEs on the book "Stochastic equations in infinite dimensions" by da Prato, Zabczyk. In the book, the theory of stochastic processes with values on a Banach space $E$, and in the particular the notion of $E$-valued brownian motion, is also introduced. When the stochastic integral is constructed, it is defined only with respect to the brownian motion using a common approximation technique (it is firstly defined for elementary processes and then extended to the closure of these processes with respect to a suitable norm; after that it can be further extended by localization).
However, when I was first introduced to stochastic analysis, it was done following the book "Cntinuous martingales and brownian motion" by Revuz, Yor, in which the (one dimensional) stochastic integral is defined for any continuous, square integrable martingale, and in particular it is shown the key result that $I_t=\int_0^t K_sdM_s$ is the unique process such that $$ \langle I,N\rangle_t = \int_0^t K_s d\langle M,N\rangle_s$$ for any $N$ continuous, square integrable martingale, where $\langle \cdot,\cdot\rangle$ denotes the bracket process, or cross quadratic variation.
My question is: can this result be extended to the general setting of $E$-valued martingales? (obviously one needs to define what is the integration with respect to the bracket in this case first) If so, do you know any books in which this is done? I made a few researches online and I found that some books approach stochastic integration in this setting using Doolean measures, which I don't know, so I'd prefer books that avoid using that kind of tool or introduce it softly before using it.
