The Wikipedia page on vector space frames gives a construction to find a dual frame for a given frame. Specifically, given a set of vectors ${ e_k }$ in a Hilbert space $\mathcal{H}$ such that for all $v\in\mathcal{H}$ and some $ A \leq B<\infty$ we have $$A||v||^2\leq\sum_k|\langle e_k|v\rangle|^2\leq B ||v||^2,$$ the construction gives (among other things) a set of linear functionals $\phi_k:\mathcal{H}\rightarrow\mathbb{C}$ such that for all $v\in\mathcal{H}$ $$v=\sum_k e_k \cdot\phi_k(v).$$
My question is, how unique are these functionals? (or alternatively, their dual images in $\mathcal{H}$.) The construction gives a natural way to find these dual images, which depends on the inverse of the map $S(v)=\sum_k\langle v|e_k\rangle e_k$ and thus on the inner product used, but that doesn't mean the corresponding functionals will depend on the inner product.
If I have, as in one standard example, three noncollinear vectors in a two-dimensional space, then I would naively expect to have "one real degree of freedom" in choosing the coordinates. How does this show up? If there are other sets of functionals, what do their dual images in $\mathcal{H}$ look like? Is the one picked by $S$ special? if so, how? Does any of this depend on the (in)finiteness of the space dimension?
