Total set of functions in $L^2(\Omega)$

Question

Are the sets of functions $\{e^{\int_0^T h(s)dB_s -\frac{1}{2}\int_0^T h(s)^2 ds}\}$ and $\{e^{\int_0^T h(s)dB_s}\}$ total in $L^2(\Omega)$? What is the difference? What should one use to prove weak convergence in $L^2(\Omega)$?

Thanks in advance! :)