I have found this statement in some old lecture notes on interpolation in my lab.
Let $\mathcal{P}_{n}(I)$ be the vector space of polynomials over some open interval $I\subset\mathbb{R}$.
Suppose some set of $m$ linear functionals in the dual space of $\mathcal{L}^2(I)$ has the property that a polynomial $p\in\mathcal{P}_{m-1}(I)$ vanishes everywhere if and only if each functional maps $p$ to 0.
(1) Is there a name for this property?
The notes say this property "guarantees the linear independence of the functionals over $\mathcal{P}_{m-1}(I)$."
What I think it means by this is the following.
If these functionals are evaluation functionals at distinct points (I guess of a representative element of each equivalence class in $\mathcal{L}^2$), this makes sense. Let's call these points $x_0, \ldots, x_{m-1}$. Using the basis $\{1, x, \ldots, x^{m-1}\}$ for the polynomials, we see that we can recover a polynomial simply by evaluating this set of functionals on it.
We can recover the $c_i$ from the following equation because the matrix on the left is Vandermonde. $\begin{pmatrix}1 & x_0 & x_0^2 & \ldots & x_0^{m-1}\\ 1 & x_1 & x_1^2 & \ldots & x_1^{m-1}\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & x_{m-1} & x_{m-1}^2 & \ldots & x_{m-1}^{m-1}\end{pmatrix} \begin{pmatrix}c_0\\c_1\\ \vdots \\ c_{m-1}\end{pmatrix} = \begin{pmatrix}p(x_0)\\p(x_1)\\\vdots \\ p(x_{m-1})\end{pmatrix}$
If some of the $x_i$ are redundant, we could use the functional that does evaluation at a derivative, and still get a sufficiently rich set of points from which we can recover an interpolating polynomial.
Given some polynomial in the $m$-dimensional space $p\in\mathcal{P}_{m-1}$ and our set of functionals $\{f_0, \ldots, f_{m-1}\}$, we can consider the mapping $f: \mathcal{P}_{m-1} \to \mathbb{R}^{m}$ defined by $fp = (f_0p, \ldots, f_{m-1}p)$. If the functionals satisfy the property, the null space of $f$ is simply the zero polynomial and so the map is one-to-one and onto. Thus, given a vector in $\mathbb{R}^m$ of "functional evaluations" one can recover the polynomial.
Yes, a collection $L$ of linear functional on a vector space is "separating" if, given vectors $v\not=w$, there is a functional in $L$ such that $\lambda(v)\not=\lambda(w)$.
Issues/algorithms about recovering $v$ from all values $\lambda(v)$ depend on the situation, but the separating property does imply that $v\to \{\lambda(v):\lambda\in L\}$ is injective.