Exercise 2.A.16

Proof. Let $C_{ℝ}[0,1]$ denote the real vector space of all continuous real-valued functions on the interval $[0,1]$. Since every polynomial is continuous on $[0,1]$, it follows that $P(ℝ)$ is a subspace of $C_{ℝ}[0,1]$. However, $P(ℝ)$ is infinite-dimensional (by 2.16). Thus 2.26 now implies that $C_{ℝ}[0,1]$ is infinite-dimensional. □