I met a problem when I read content on multidimensional stochastic integral content. The book says that $dB_i(t)dB_j(t)=\delta_{ij}dt$ where $\delta_{ij}=1$ if $i=j$ and $\delta_{ij}=0$ if $i \neq j$. It seems intuitive to me but I know this is not rigorous. So how would I calculate the $\mathbb{E}\left( \int_t^{t+\epsilon} \sigma_{ik}(s,X_s)dB_k(s) \int_t^{t+\epsilon} \sigma_{jl}(s,X_s)dB_l(s) \right)$?
Here $B_k(t)$ and $B_l(t)$ are two different components of $B(t)\in\mathbb{R}^n$. So does it mean that the two integrals $\int_t^{t+\epsilon} \sigma_{ik}(s,X_s)dB_k(s)$ and $\int_t^{t+\epsilon} \sigma_{jl}(s,X_s)dB_l(s)$ are independent? If so, then is it true that \begin{align} &\mathbb{E}\left( \int_t^{t+\epsilon} \sigma_{ik}(s,X_s)dB_k(s) \int_t^{t+\epsilon} \sigma_{jl}(s,X_s)dB_l(s) \right) \\ &= \mathbb{E}\left( \int_t^{t+\epsilon} \sigma_{ik}(s,X_s)dB_k(s) \right)\mathbb{E}\left( \int_t^{t+\epsilon} \sigma_{jl}(s,X_s)dB_l(s) \right) = 0 ? \end{align}
By Ito isometry, we have
\begin{align} &\mathbb{E}\left( \int_t^{t+\epsilon} \sigma_{ik}(s,X_s)dB_k(s) \int_t^{t+\epsilon} \sigma_{jl}(s,X_s)dB_l(s) \right) \\ = & \mathbb{E}\left( \int_t^{t+\epsilon} \sigma_{ik}(s,X_s) \sigma_{jl}(s,X_s)d[B_k,B_l](s) \right) \\ = & 0 \end{align} since $[B_k,B_l](s)=0$ if $B_k$ and $B_l$ are independent.