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In a proof of random matrix theory, the author makes use of the following equivalence:

\begin{equation} \inf_{v\in V(r)}\lVert Xv \rVert_2 = \inf_{v\in V(r)} \sup_{u \in S^{n-1}}u^TXv \end{equation} where $X$ is $p X n$ random matrix whose rows $X_i$ follows the distribution $X_i \sim \mathcal{N}(0,K)$ $\forall i \in \{1,2,\ldots n\}$ with $S^{n-1} = \{ u \in \mathbb{R}^n | \lVert u \rVert = 1\}$ and $V(r) = \{ v \in \mathbb{R}^p | \lVert K^{1/2}v\rVert_2 = 1, \lVert v\rVert \leq r \}$.

Can someone please help understand this equivalence?

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    $\begingroup$ Does this help? Also, the dual of the $\ell_2$ norm is the $\ell_2$ norm . $\endgroup$ – Vincent Guillemot May 21 at 14:13
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    $\begingroup$ There is a lot of irrelevant distracting material that hides a simpler statement: focus on vectors $v,$ matrices $X,$ and consider various equivalent definitions of the norm (as suggested by @Vincent). Ignore the random variables and the $\inf$ operations. $\endgroup$ – whuber May 21 at 14:14
  • $\begingroup$ @VincentGuillemot Sorry, I'm not well-versed with with these spaces. Could you please elaborate a bit? $\endgroup$ – Akshay Bansal May 21 at 14:35
  • $\begingroup$ @VincentGuillemot Okay, I see it now. Sorry, my bad. It's a simple application of Cauchy-Schwarz $\endgroup$ – Akshay Bansal May 21 at 14:45
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    $\begingroup$ That's an excellent way to see it! $\endgroup$ – Vincent Guillemot May 21 at 15:18

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