## Set of Real Polynomial Functions

This problem was given as an exercise in a Functional Analysis lecture. I’ve tidied up my old solution for the problem, and here it is. Definitions Definition 1. (Sub-vector Space) Suppose that \(V\) and \(W\) are two vector spaces and \(0_V \in W \subseteq V\). W is a sub-vector space of V iff $$\forall w_1, w_2 \in W \land v \in V : v(w_1 + w_2) \in W$$ Definition 2. (Metric Space) Let \(\empty \neq X\) be a set and \(d : X \times X \mapsto \R\) be a function....