On Susskind book, exercise 3.1 say to prove the following:
"If a vector space is $N$-dimensional, an orthonormal basis of $N$ vectors can be constructed from the eigenvectors of a Hermitian operator."
Susskind wrote that the proof is easy.
From the book I understand that with two unequal eigenvalues of a Hermitian operator, then the corresponding eigenvectors are orthogonal. Even if the two eigenvalues are equal, the corresponding eigenvectors can be chosen to be orthogonal.
But I do not undesrtand how to prove if the space is N-dimensional, there will be N orthonormal eigenvectors.
Please Help.