Bounded synthesis of square-summable Hilbert columns with exact adjoint energy and uniqueness

Let $H$ be a complete complex Hilbert space and $(v_i)_{i\in I}$ a family with $\sum_i\|v_i\|^2<\infty$. There is a unique bounded complex-linear map $S:\ell^2(I,\mathbb C)\to H$ having these basis columns. It is given by the absolutely convergent synthesis series $Su=\sum_i u_i v_i$, satisfies $\|S\|^2\le\sum_i\|v_i\|^2$, and its adjoint obeys

The Hilbert inner product is conjugate-linear in its first argument. All sums are unconditional, and no countability assumption on $I$ is required.

This is the functional-analytic construction used in the Connes–Weil Green-column frontier: it derives bounded synthesis, exact adjoint norms, and uniqueness from square-summable supplied columns. Its hypothesis is square summability of those columns; it does not construct a physical Green metric, prove a zeta-zero counting bound, or prove RH. Identification with an existing native operator requires the same coefficient carrier and the same basis columns.