Finite-dimensional well-definedness of the filtered γ₂ program

If A_{xx}≠0, the factorization equation at (x,x) has right side zero because every Δ_i(x,x)=0, so A is infeasible. If A has zero diagonal, assign each off-diagonal entry A_{xy} to the least i with x_i≠y_i, forming matrices A^{(i)} supported on Δ_i and satisfying A=Σ_i A^{(i)}. For d=|D|, put u_{x,i}=(\overline{A^{(i)}{x1}},…,\overline{A^{(i)}{xd}}) and v_{y,i}=e_y in C^d; then ⟨u_{x,i}|v_{y,i}⟩D=A^{(i)}{xy}. This is a finite feasible factorization.

For any feasible factorization in finite dimension, take, for each i, the span of the d vectors $u_{x,i}$ and d vectors $v_{y,i}$. It has dimension at most $2d$; by [F4] choose an orthonormal basis and map its vectors to their coordinates, padding with zero coordinates in ${\mathbb{C}}^{2d}$. This preserves every inner product and norm. Thus every feasible candidate has a representation in one fixed finite product of coordinate spaces ${\mathbb{C}}^{2d}$.

By step 1.1 choose one feasible family and let $K$ be its finite cost. Use step 2.1 to represent feasible families in a fixed finite product of copies of ${\mathbb{C}}^{2d}$, viewed as a positive-dimensional real Euclidean space; $d\ge 1$ by [F7]. The set of coordinate families satisfying the factorization equations and having cost at most $K$ is nonempty, closed, and bounded: the equations and cost are continuous, and each vector norm is at most $\sqrt{K}$. By [F5] this set is compact. Its continuous cost function attains a minimum by [F6]. Any feasible family with lower cost would also belong to this set, so the minimum equals the global infimum and is attained. Thus every feasible $A$ has an attained finite value, while nonzero diagonal is exactly the infeasible case.