B e_j \rangle\end{aligned}$$ is also a separable Hilbert space. In $\mathcal{S}(\mathbb{R}^d)$, we have equivalently $$\begin{aligned}
\langle A, B \rangle_F &= \mbox{tr} \left( A^T B \right),\end{aligned}$$ where $A^T$ is the transpose matrix of $A$. The induced norm is the well known Frobenius norm (also called Hilbert-Schmidt norm) and is denoted by ${\left\| . \right\|}_F .$ When $X$ has finite second order moments, with expectation ${{\mathbb{E}}\left[X\right]}=\mu$, the covariance matrix of $X$, ${{\mathbb{E}}\left[(X-\mu)(X-\mu)^T\right]}$ can be defined as the minimum argument, over all the elements belonging to $\mathcal{S}(H)$, of the functional $G_{\mu,2} : \mathcal{S}(H) \to \mathbb{R}$, $$G_{\mu,2}(\Gamma) = {{\mathbb{E}}\left[ {\left\| (X-\mu)(X-\mu)^T - \Gamma \right\|}^2_F - {\left\| (X-\mu)(X-\mu)^T \right\|}_F^2\right]}.$$ Note that in general Hilbert spaces with inner product $\langle ., . \rangle$, operator $(X-\mu)(X-\mu)^T$ should be understood as the operator $u \in H \mapsto \langle u, X-\mu \rangle (X-\mu)$. The MCM is obtained by removing the squares in previous function in order to get a more robust indicator of “covariation”. For $\alpha \in H$, define $G_\alpha : \mathcal{S}(H) \to \mathbb{R}$ by $$\begin{aligned}
G_\alpha (V) &:= {{\mathbb{E}}\left[ {\left\| (X-\alpha)(X-\alpha)^T - V \right\|}_F - {\left\| (X-\alpha)(X-\alpha)^T \right\|}_F\right]} . 
\label{def:popriskcov}\end{aligned}$$ The median covariation matrix, denoted by $\Gamma_m$, is defined as the minimizer of $G_m(V)$ over all elements $V 