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.
Edit: this answer was given before an edit to the question and does not validly answer the new question.
You can define such a space from the ground up: Let $V$ be the vector space of finite (i.e. eventually always zero) linear combinations of a countable set $\{e_i | i \in \mathbb{N}\}$. Then define the inner product as $\langle \sum x_i e_i , \sum y_i e_i \rangle = \sum x_i \bar{y_i}$, where this sum always exists because eventually both the $x_i$ and the $y_i$ become $0$.
In this vector space, the basis vectors $e_i$ are orthonormal.