Space of Polynomials over a Field form a Vector Subspace of Function Vector Space

Last updated Nov 1, 2022

Let $F$ be a Field and denote $$P[F] = {x \mapsto \sum\limits_{k=0}^{n} c_{k}x^{k} : n \in \mathbb{Z}{\geq 0}; c{0}, \dots, c_{n} \in F }$$ the Set of all Polynomials over $F$. Then $P[F]$ forms a Vector Subspace of the space $F^{F}$.

We know from Functions to a Field form a Vector Space that $F^{F}$ forms a Vector Space. By A Subset of a Vector Space is a Subspace iff it is closed under scaling and addition, it suffices to check we are closed under scaling and addition.

Suppose $c \in F$ and $p, r \in P[F]$. Then there exist $n, m \in \mathbb{Z}{\geq 0}$ so that $$\begin{align*} &p = c{0} + c_{1} x + \cdots + c_{n} x^{n}\\ &r = d_{0} + d_{1} x + \cdots + d_{m} x^{m}\\ \end{align*}$$ Without Loss of Generality we can take $n \geq m$ and then rewrite $$r = d_{0} + d_{1} x + \cdots + d_{m} x^{m} + d_{m+1} x^{m+1} + \cdots d_{n} x^{n},$$ taking $d_{m+1}, \dots, d_{n} = 0$. Then we get that $$\begin{align*} cp + r &= \sum\limits_{k=0}^{n}c c_{k}x^{k} + \sum\limits_{k=0}^{n} d_{k}x^{k}\\ &=\sum\limits_{k=0}^{n}(c c_{k} + d_{k})x^{k} \in P[F] \end{align*}$$

$\blacksquare$

1. This also establishes that Polynomials over a Field form a Vector Space.