Gram-Schmidt Orthonormalisation

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space, let $m$ be a natural number, and let $b\in V^{m}$ be an $m$-tuple in $V$ that is linearly independent. For $k\in[m]$, with $[k]$ the initial segment determined by $k$, write $x|_{[k]}$ for the restriction of a tuple $x$ to $[k]$.

Then there is an orthonormal tuple $e\in V^{m}$ with

$$\operatorname{span}\bigl(e|_{[k]}\bigr)=\operatorname{span}\bigl(b|_{[k]}\bigr)\qquad\text{for every }k\in[m],$$

the span being that of a finite tuple.