Derivation — Proof of the Spectral Theorem by Induction on Invariant Subspaces

This is the technical heart of the unit's most important theorem. The previous note stated the real spectral theorem's substantial direction — self-adjoint $\Rightarrow$ orthonormal eigenbasis — without proof. This note carries that proof: peel off one eigenvector, show the space orthogonal to it is itself invariant and the operator restricted to it is still self-adjoint, then recurse. The technique — invariant-subspace induction — is new to the unit and reappears throughout the rest of Stage 2 and Stage 3.

Learning objectives. By the end of this note you should be able to: