I am currently working on a problem from High-Dimensional Statistics by Martin Wainwright, where the goal is to bound the expectation of the maximum singular value of an $n \times d$ matrix with sub-Gaussian $\sigma$ entries. Let $M_{n, d}(1)$ denote the set of rank-$1$ matrices with unit Frobenius norm. In part of the proof, the author uses the inequality
$$ \mathbb{E}\left[\sup_{\substack{\Gamma, \Gamma' \in M_{n,d}(1) \\ ||\Gamma - \Gamma'||_{F} < \delta}}\langle\Gamma - \Gamma', W\rangle\right] \leq \sqrt{2}\delta\mathbb{E}[||W||_2], $$
where the inner product is the Frobenius inner product (i.e. the inner product on the vectorized matrices). Could anyone give hints as to how I could approach this problem? I have tried applying Cauchy-Schwarz to the vectorized matrices and for every row of $W$, but neither of those have been very fruitful. I'm not quite sure how to leverage the fact that $\Gamma$ and $\Gamma'$ are rank-$1$ matrices.
As it turns out, one can apply a version of Holder's inequality for the Frobenius inner product; in particular, we have that
$$ \langle \Gamma - \Gamma', W\rangle \leq \lVert\Gamma - \Gamma'\rVert_1\lVert W\rVert_\infty $$
where $\lVert\cdot\rVert_p$ is the Schatten $p$-norm. Noting that the Schatten $\infty$-norm is the operator norm of $W$ and that $\lVert\Gamma - \Gamma'\rVert_1 \leq \sqrt{2}\lVert \Gamma - \Gamma'\rVert_F$ (since the difference is at most rank $2$), we obtain the desired result.