Infinite dimensional Gram-Schmidt

Question

Given an infinite dimensional inner product space $(V,\langle \cdot,\cdot \rangle)$, with a countable Hamel basis, is it always possible to perform the Gram–Schmidt process and produce an orthonormal basis for
$(V,\langle \cdot,\cdot \rangle)$? (To be precise, by orthonormal basis I mean a Hamel basis $\{e_i\}_{i \in \mathbb{N}}$ such that $\langle e_i,e_j \rangle) = \delta_{ij}$, for all $i,j$?

Answer 1

Yes and this is quite straightforward. Let $(v_n)$ be a Hamel basis. Having found an orthonormal basis $e_1,e_2,...,e_n$ for the span of $v_1,v_2,..,v_n$ we can always find coefficients $c_j$ such that $e_{n+1}=\sum c_je_j+c_{n+1} v_{n+1}$ has norm $1$ and is orthogonal to $e_i: i \leq n$.