In one of Talagrand's paper on isoperimetric inequalities, it says
"... Consider independent Bernoulli random variables ($P(\epsilon_i = 1) = P(\epsilon_i = -1) = 1/2$) and vectors $(x_{ij})_{i,j\leq n}$ is a Banach space $W$. We assume $x_{ii} = 0$ for all $i \leq n$, and that $x_{ij} = x_{ji}$. We denote by $W_1^*$ the unit ball of $W$, and consider the numbers $$ U = \sup\left\{\sum_{i,j\leq n} \alpha_i \beta_j x^* (x_{ij});\ x^* \in W_1^*, \sum_{i\leq n}\alpha_i^2 = 1, \sum_{j \leq n}\beta_j^2 = 1\right\} $$ Thus $U$ is the supremum of the operator norms of the matrices $(x^*(x_{ij}))_{i, j}$. ..."
I am confused on the last sentence about $U$. Observe $U$ is obtained by taking supremum on a set of matrices by definition, since $x^*$ is a matrix. How can it be a number? Am I missing something?
$x^*$ is in $W_1^*$ so doesn't denote a matrix, it denotes a linear functional $x^*:W\to \mathbb R.$ The notation $(x^*(x_{ij}))_{i,j}$ does denote a matrix - it's $x^*$ applied entrywise. For each fixed $x^*,$ the quantity
$$ \sup\left\{\sum_{i,j\leq n} \alpha_i \beta_j x^*(x_{ij}); \sum_{i\leq n}\alpha_i^2 = 1, \sum_{j \leq n}\beta_j^2 = 1\right\} $$
is the $\ell_2$ operator norm (aka spectral norm) of the matrix $(x^*(x_{ij}))_{i,j}.$ Taking the supremum over $x^*\in W_1^*$ gives $U.$