Existence of the $\Omega$ set in "Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization" In the proof of Lemma 4.3 in [1], they claim the following:
Let $U$ be a subspace of $\mathbb{R}^{m\times n}$ with dim$(U)=d$ and let $\delta>0$. Then, there exists a set $\Omega\subset\mathbb{R}^{m\times n}$ wiht at most $(12/\delta)^d$ elements such that for every $X\in U$ with $\lVert X\rVert_F\leq 1$ there exists a $Q\in\Omega$ such that $\lVert X-Q\rVert_F \leq \delta/4$.
I don't see the reason why this is true.
References
[1] Recht, B., Fazel, M., & Parrilo, P. A. (2010). Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM review, 52(3), 471-501.
 A: I know the following proof. The space $U$ endowed with the Frobenius_norm is isometric to a Euclidean space $\Bbb R^d$ and $V=\{X\in U: \|X\|_F\le 1\}$, that is $V$ is a unit ball of a $U$. Given a number $\varepsilon>0$, a subset $\Theta$ of $V$ is $\varepsilon$-separated, if $\|Q-Q'\|_F\ge\varepsilon$ for each distinct $Q,Q’\in\Theta$. Let $\Omega$ be a maximal $\delta/4$-separated subset of $V$. Then for each $X\in V$ there exists  $Q\in\Omega$ such that $\lVert X-Q\rVert_F \leq \delta/4$. On the other hand, for each $Q\in\Omega$ let $B(Q)\subset U $ be an open ball of radius $\delta/8$ centered at $Q$. Then for each distinct $Q,Q’\in\Omega$, balls $B(Q)$ and $B(Q’)$ are disjoint and are contained in an open ball $B$ of radius $1+\delta/8$ centered at zero. Comparing the $d$-dimensional volumes of $B$ and $B(Q)$ for $Q\in\Omega$, wee see that 
$|\Omega|(\delta/8)^d\le (1/2+\delta/8)^d,$
that is 
$|\Omega|\le \left(1+\frac  4{\delta}\right)^d$.
So $|\Omega|\le \left(\frac  {12}{\delta}\right)^d$ for $\delta\le \tfrac 18$. 
