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I am going through Sheldon Axler - Linear Algebra Done right. The book States theComplex Spectral Theoremas :

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, theReal Spectral TheoremStates 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.

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# The Spectral Theorem in Complex and Real Inner Product Space

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