Hi I am going through Sheldon Axler - Linear Algebra Done right. The book States the Complex Spectral Theorem as : Suppose that V is a complex inner product space and T is in L(V,V). Then V has an orthonormal basis consisting of eigen vectors of T if and only if T is normal. The proof of this theorem seems fine. It uses the property that ||Tv|| = ||T*v|| for a normal operator T, where T* is the adjoint of T. However, the Real Spectral Theorem States that V has an orthonormal basis consisting of eigen vectors of T if and if only if T is self adjoint. My Doubt : Why does Real Spectral Theorem take into account only self adjoint operators as a necessary condition despite the fact that an operator can be normal and still not self adjoint. When it's normal, the property ||Tv|| = ||T*v|| should be still valid for real inner product space which leads to the desired result. Would be great if somebody could give me an insight. Thanks.