I would like to ask whether or not, my reasoning is correct.
This is probably a dumb question, but I haven't done any functional analysis course, and am kind of worried if the following reasoning (which seems to me as intuitive) is all right.
I am trying to solve a problem about some particular set of random variables $\ \mathcal{S}, \ $ $$\mathcal{S}\subset L_{2}(\Omega):= \{X: \mathbb{E}X^{2}<\infty\}.$$ I have proved that $S$ is a subset of $$\hat{\mathcal{S}}=\{X\in L_{2}(\Omega): \mathbb{E}|X-X_{0}|^{2}=1\},$$ for some r.v. $\ X_{0}\in L_2(\Omega). \ $ I want to find a bound/calculate $$\sup_{X_1,X_2,\cdots,X_n \in \mathcal{S}}\sum_{i,j}\mathbb{E}|X_i-X_j|^2.$$ And now, what i would like to say is:
$X_1,X_2,\cdots,X_n \ $ are $\ n \ $ points in Hilbert space $\ L_{2}(\Omega), \ $ so they must lie on some $\ n-1 \ $ dimensional subspace $\ H \ $. This subspace is (I believe) isometric to the space $\ \mathbb{R}^{n-1} \ $. Let $\ x_{1},\cdots, x_{n} \ $ be corresponding set of points in $\mathbb{R}^{n-1}$. Then $\ x_1,\cdots, x_n$ still lies on some sphere $\hat{\mathcal{S}}'$ with radius $1$. Finally $$\sum_{i,j}\mathbb{E}|X_i-X_j|^2=\sum_{i,j}||x_i-x_j||^2,$$ so $$\sup_{X_1,X_2,\cdots,X_n \in \mathcal{S}}\sum_{i,j}\mathbb{E}|X_i-X_j|^2\le \sup_{x_1,x_2,\cdots,x_n \in \hat{\mathcal{S}}'}\sum_{i,j}||x_i-x_j||^2$$ which is now an euclidean geometry problem (already solved).
Is this reasoning correct? I will be glad for any help.