I'm having trouble digesting the proof of Theorem 9.6 in Milman and Schechtman's classic book "Asymptotic theory of finite dimensional normed spaces" (pg. 54). I'm new to functional analysis, so this is surely a simple misunderstanding on my part.
Here, $M_{\|~\|}$ is denoting the average norm for points on $S^{n-1}$. The first line in the computation of $M_{\|\cdot\|}$ is confusing me:
$M_{\|\cdot\|} \approx \left(\int_{S^{n-1}} \|\sum a_i e_i\|^2 d\mu(\bar{a})\right)^{1/2}$
We're actually trying to show a lower bound here, so isn't the $\approx$ hiding a $1/\sqrt{n}$ factor? Isn't this bad for the rest of the argument? Earlier in the book, in the proof of Dvoretzky's Theorem, $M_{\|\cdot\|}$ is given as $\int_{S^{n-1}} \|\sum a_i e_i\| d\mu(\bar{a})$, which is what I'd have expected here also.