one can define the conditional expectation of a random variable $X$ on $\mathbb{R}^n$ in an axiomatic way, without relying on any norm by being the $\mathcal{G}$-measurable random variable $Y$ which fulfills $$\mathbb{E}[Y 1_A] = \mathbb{E}[X 1_A]$$ for any $A \in \mathcal{G}$.
However, one can also define it as the $L^2(\mathbb{R}^n, \mathcal{F}, \mathbb{P})$ projection onto all $\mathcal{G}$-measurable random variables, where $\mathbb{P}$ is the distribution of $X$. But this projection is defined by minimizing $$\mathbb{E}[\|X - Y\|^2]$$ and therefore depends on the choice of norm $\|\cdot\|$ on $\mathbb{R}^n$. How can that be?
Or to phrase it more generally, if I have the conditional expectation on some topological space, where I do not have such a natural norm as in $\mathbb{R}^n$, with respect to which norm will it be the $L^2$ projection?