# Definition # Vector Space The set of all $\mathbb{F}$-coefficient ($\mathbb{F}$ is a [[field]], e.g., $\mathbb{C}$) polynomial functions on $\mathbb{R}^n$ of fixed degree $l$ of the form $ \prod_{i = 1}^{n} x_{i}^{l_i} $ such that $ \sum_{i = 1}^{n} l_i = l $ are denoted $P_l(\mathbb{R}^n)$, and they form a [[vector space]] over $\mathbb{F}$. As a simple example, let $n = 3$, and the polynomials then take the form $x^i y^j z^k$ with $i + j + k = l$. This is the space $P_l(\mathbb{R}^3)$.