Given is the following lemma
If the random variables $X_1, X_2 \in L^2([0,1])$ are independent, $E[\|X_i\|_{L^2}^2] < \infty$ with $i = 1,2$ and $E[X_1] = 0$, then $$E[ \langle X_1, X_2 \rangle_{L^2} ] = E \left[\int\limits_0^1 X_1X_2 \right] = 0$$
My idea for the proof is: $E[\langle X_1, X_2 \rangle_{L^2}] = E[\langle X_2, X_1 \rangle_{L^2}] = \langle X_2, E[X_1] \rangle_{L^2} = \langle X_2, 0 \rangle_{L^2} = 0$.
Now I asked a friend of mine who told me that the "basic idea" behind this is true, but that the above expression is not defined. I have to work with conditional expectation and iterated expectations. But I don't get it. Why do I need these two results and what is the correct proof then? Thanks!