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vish_maths

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Hi

I am going through Sheldon Axler - Linear Algebra Done right. The book States the

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

Would be great if somebody could give me an insight. Thanks.

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.

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