Let $$X_T=\int_0^T F_tdt+\int_0^T G_tdW_t.$$ Why $$\left<X\right>_T=\int_0^T G_t^2dt\ \ ?$$ I recall that $\left<X\right>_T$ is the quadratic variation.
Attempts
I know that if $M$ is a martingale, then $\left<M_t\right>$ is the unique increasing adapted process s.t. $M_t^2=N_t+\left<M_t\right>$ where $N_t$ is a martingale. So, I guess that $\left<X\right>_t$ is a "notation" for $\left<\int_0^{\cdot }G_sdW_s\right>_t$.
Question : Am I right ? (because in the sense of my definition, since $X$ is not a martingale, I don't see what would be $\left<X\right>$ according to my definition).