I would like to show that if $Y\in\Re$ is a random real vector such that $\|Y\|≤ \bar Y$ and $X \in \mathcal H$ where $\mathcal H$ is a Hilbert space (whose field is $\Re$), then $$ \|\operatorname{Cov} (Y,X)\| = \|E(YX^*)\| ≤ \bar Y \|X\|_∞. $$ The first idea is the following: if $X∈ \Re^p$, then I think that $$ \|\operatorname{Cov} (Y,X)\| = \|E(YX^T)\| ≤ \sqrt{\operatorname{Var} (Y)}\max_{i=1,\ldots,p}\sqrt{\operatorname{Var} (X_i)} $$ which would yield the first equation based on the fact that if $|Z|≤\nu$, then $\operatorname{Var} (Z) ≤ ν^2$.
The second idea (which yields the desired inequality in the real case when $X_i$ are independent one to each other) is to consider the fact that $X$ has limited total information upon $Y$.