Theorem about coordinates of a subspace in an inner product space

Question

Theorem: Let $B = {v_1,v_2,...,v_n}$ be an orthonormal basis of an inner product space V and let $w∈V$.Then the coordinates $a_1,a_2,...,a_n$ of $w$ in the basis B are given by $a_i =⟨w,v_i⟩$.

Do coordinates in this case mean scalars on subspace w?

Answer 1

Given a basis $\mathcal{B}=(v_1,\ldots, v_n)$ for a vector space $V$ over a field $k$ $(=\mathbb{R},\mathbb{C}$ for instance), and given an element $w\in V$, we say that the coordinates for $w$ with respect to $\mathcal{B}$ are the unique $a_1,\ldots, a_n\in k$ such that $w=a_1v_1+\cdots+a_nv_n$.

Of course, the facts that such $a_1,\ldots, a_n$ exist and are unique follow from the definition of a basis.