Proposition 3.1

Let $H$ be a complete complex inner product space, equipped with its normed additive commutative group structure, whose dimension as a complex vector space, expressed as the cardinal $\operatorname{rank}_{\mathbb C}H$, is strictly greater than $1$; infinite dimension is allowed. Suppose that to each closed complex linear subspace $U\subseteq H$ a type $S(U)$ is assigned, with a restriction map $\rho_{U,V}:S(V)\to S(U)$ for every inclusion $U\subseteq V$, such that $\rho_{U,U}$ is the identity and $\rho_{U,V}\circ\rho_{V,W}=\rho_{U,W}$ for nested subspaces. Assume that for every $U$, every set $\mathcal C$ of closed subspaces whose supremum—its closed linear span—is $U$, and every choice $s_V\in S(V)$ for $V\in\mathcal C$, agreement of the restrictions of $s_V$ and $s_W$ to $V\cap W$ whenever $V\cap W\ne\{0\}$ implies the existence of exactly one $s\in S(U)$ restricting to every $s_V$. This compatibility requirement imposes no condition on pairs with zero intersection, and the gluing assumption includes the empty cover of the zero subspace. If, in addition, every $S(U)$ is nonempty, then for every closed subspace $U$ there exists exactly one element of $S(U)$. No algebraic structure on these types of sections is assumed.