Let $H$ be a real separable Hilbert space. Suppose $\mu\sim N(0,S)$ on $H$, where $S$ is a positive, symmetric, of trace class operator on $H$. If $\lambda$ is a measure on $H$ with the Radon-Nikodym derivative $$\frac{d\lambda}{d\mu}(x)=\exp[(c,x)-\frac{1}{2}(Sc,c)]$$ for some $c\in H$. Prove that $\lambda\sim N(Sc,S)$.
My attempt: I tried to calculate the characteristic function $$E_{\lambda}[e^{i(x,y)}]=\int_H e^{i(x,y)}\lambda(dx)=\int_H e^{i(x,y)}e^{(c,x)-\frac{1}{2}(Sc,c)}\mu(dx)$$ and then I was stuck there. I thought I should do some kind of change of variables and then use the property that $\mu$ is Gaussian. Could anyone give me some hints?
Thanks!