**Prove if the following statement is true.**

Let V be a finite dimensional C-vector space with inner product and T a diagonalizable linear operator on V. Then there is a basis of eigenvectors of T for V. Applying the Gram-Schmidt orthogonalization process to this basis and then normalizing, an orthonormal basis of eigenvectors for V is obtained. By the Complex Spectral Theorem, T is normal.

**Attempt:**

T a diagonalizable linear operator on V if and only if T has a basis of n eigenvectors, in which case the diagonal entries are the eigenvalues for those eigenvectors. Let $\beta$ be that basis.

By Gram-Schmidt, ${\beta}'$ is an orthonormal basis of V.

Since T is diagonalizable, $[T]_{{\beta}'}$ is diagonalizable.

There exists an orthonormal basis ${\beta}'$ of V such that $[T]_{{\beta}'}$ is diagonalizable if and only if T is normal.

Therefore, the statement is true.

Am I wrong?