I am currently reading Foucart and Rauhut's A Mathematical Introduction to Compressive Sensing, and I have some issues understanding some of the notation with respect to the $\mathcal{l}_1$-coherence.
Let $A \in \mathbb{C}^{mxN}$ be a matrix with $\mathcal{l}_2$-normalized columns $a_1,...,a_N$. The $\mathcal{l}_1$-coherence function $\mu_1$ of the matrix $A$ is defined for $s \in [N - 1]$ by
$\mu_1(s) = \max_{i\in [N]} \max{\{ \sum_{j \in S} | <a_i,a_j> |, S \subset [N], \mathrm{card}(S)=s, i \not\in S \}}$
I just cannot wrap my head around how to interpret the double maximization here. Because what does the right max term maximize over without the $i$-index, how would the sum of inner products be maximized without maximizing over $i$?
Or do I read the notation from the wrong direction?
You could define for each $i$ and $s$ the set $$ {\cal F}(i,s) = \bigl\{ S \mid S \subset [N], \operatorname{Card}(S) = s \text{ and } i\not\in S \bigr\} $$ in order to rewrite your formula $$ \mu_1(s) = \max_{i\in [N]} \max{\left\{ \sum_{j \in S} | \langle a_i,a_j \rangle |, S \subset [N], \mathrm{card}(S)=s, i \not\in S \right\}} $$ as follows $$ \mu_1(s) = \max_{i\in [N]} \max_{S \in {\cal F}(i,s)}{\sum_{j \in S} | \langle a_i,a_j \rangle | } $$ which should make clear how the two successive max are computed.