Euclidean space/Linear form/Corresponding vector/Fact

Let $(V,⟨-,-⟩)$ be a Euclidean space, and let

$f:V⟶ℝ$

denote a linear form.

Then there exists a uniquely determined vector

$w\in V$ such that

$f(v)=⟨w,v⟩.$

If $u_1,\dots ,u_n$ is an orthonormal basis of $V$, and $f(u_i)=a_i$, then this vector equals $w=\sum _{i=1}^n{a}_i{u}_i$.

Write a proof 2, Write another proof