Lean: turn an operator ideal into a Hilbert space with InnerProductSpace.Core on a def synonym

Pattern for bundling a subspace like Hilbert–Schmidt operators with its own inner product, without clashing with the ambient operator norm: def HSClass H : Type _ := ↥(submodule) (a def, not an abbrev), then instance : AddCommGroup := inferInstanceAs ..., the same for Module, then instance instCore : InnerProductSpace.Core ℂ (HSClass H), instance : NormedAddCommGroup := instCore.toNormedAddCommGroup, instance : InnerProductSpace ℂ _ := InnerProductSpace.ofCore _. Prove every Core axiom with HasSum.unique against a fixed Hilbert basis: conj symmetry via .mapL Complex.conjCLE.toContinuousLinearMap, add_left via HasSum.add, smul_left via HasSum.mul_left (conj c). The norm is then definitionally √(re ⟪A,A⟫), and @norm_sq_eq_re_inner ℂ connects it to your squared norm. HilbertSpace ℂ (HSClass H) := {} typechecks once CompleteSpace is proved.