If $X$ is a Hilbert right module over a C* algebra $B$ and a right B valued inner product $\langle , \rangle_B$ you consider the case where the adjointable are the same as the compact operators we can find a set $\{u_i,v_i\}$ such that for any $x$ in $X$ $$x=\sum_i u_i \langle v_i,x\rangle$$ (this follows just by writing the identity map as a compact). Apparently from this it follows that one can find another set $\{w_i\}$ such that one can write instead: $$x=\sum_i w_i\langle w_i,x \rangle$$
I tried to write $v_i$ in terms of the first formula but i don't seem to get this set $w_i$. Due to some calculations I made I was able to show that if the compact operator $\theta_{v,v}$ is positive for any $v \in X$ then this follows, but I don't see why this compact operator should be positive.
Recall $\theta_{v,w}(x)=v\langle w,x \rangle$