Like the classical Gram-Schmidt theorem for symplectic vector spaces, the sheaf-theoretic version (in which the coefficient algebra sheaf $𝒜$ is appropriately chosen) shows that symplectic $𝒜$-morphisms on free $𝒜$-modules of finite rank, defined on a topological space $X$, induce canonical bases (Theorem 1.1), called symplectic bases. Moreover (Theorem 2.1), if $(ℰ,\phi )$ is an $𝒜$-module (with respect to a $ℂ$-algebra sheaf $𝒜$ without zero divisors) equipped with an orthosymmetric $𝒜$-morphism, we show, like in the classical...