This question already has an answer here:
For a finite-dimensional inner product space $V$, since it has a finite basis, then we can do the Gram-Schimidt process to produce an orthonormal basis. However, the Gram-Schmidt process does not work with infinitely many vectors. So it is natural to ask, does every infinite-dimensional inner product space have an orthonormal basis? If the answer is yes, how to prove it?
PS: For "basis", I mean the Hamel basis.