There is a few thing that is not clear for me about those subjects that I would like for an explanation.
The spectral theorem: according to my understanding, it says that if I have an operator $T$ or matrix $A$ in finite dimension space, we knot that $T$ is normal $\iff$ I have orthonormal basis to $V$ of eigenvectors of T.
my questions:
$1.$ that orthonormal basis is orthonormal with respect to the standard inner product space? or it can be true for any product space?
$2.$ if the answer for 1 is for any inner product space, what guarntee me that I have unitary $P$ matrix that I can apply the unitary diagonalization?
Saying that $T$ is normal means that $TT^*=T^*T$. You should keep in mind that $T^*$ is determined from the equation $\langle T\vec{v},\vec{u}\rangle=\langle \vec{v},T^*\vec{u}\rangle$, which means that for different inner products you will get different $T^*$'s.
So the spectral theorem works in general, and does not require any specific inner product. The "secret" is that the contidion "T is normal" is the condition that makes it work for whatever inner product you choose.
This should also answer the second question since $P$ being unitary means that $P^*=P^{-1}$ which also depends on the inner product endowed on your vector space.