@kodlu Consider an orthonormal basis, represented by a matrix $U$, for a $k$-dimensional subspace of $\mathbb{R}^{Ck\log(k)}$. As $U$ has $Ck\log(k)$ rows and $\|U\|_F^2 = k$, there must exist a row of $U$, say $U_{i}$ with $\|U_i\|_2 \ge \sqrt{k/Ck\log(k)}$. So, $U \cdot (U_i^T)/\|{U_i^T}\|$ which is a unit vector in the subspace has a coordinate of value $\|U_i\|_2 \ge \sqrt{k/Ck\log(k)} = \sqrt{1/C\log(k)}$.