5 The upper potential and its barrier

The second family is carried through the construction by a strictly positive operator \(B\) on a separable Hilbert space \(H\), which must stay above a multiple of the Gram operator \(J = \int _\Omega b(x) b(x)^* \, d\mu (x)\) of Lemma 3.2, as the matrix \(A\) of Chapter 4 stays above a multiple of the identity. It is controlled by a single real number, and whether a candidate point may be used is decided by a function of that point. Nothing in the argument ties the number of points to the size of the second family: it is governed by the effective dimension alone. A finite second family is the case \(H = \mathbb {C}^{\kappa }\), where every operator below is a matrix and every trace a matrix trace.