$P_3=\{ a_0+a_1x+a_2x^2+a_3x^3: a_i\in \mathbb{R} \}$, and $C[0,1]$ is the set of all functions continuous on $[0,1]$. I need to show that $P_3\subset C[0,1]$ by exposing its basis and dimension. We know that in order to show that a subset is a subspace of a vector space we need to show closure under addition and scalar multiplication, as well as that the zero-element is in the subset. But here it appears different. I'm not completely sure if my understanding is correct, so I would appreciate if my solution is reviewed.
Solution:
Let $H$ be a basis for $C[0,1]$, then $H$ is infinite. Also, $C[0,1]$ includes all polynomial functions since they are continuous on $\mathbb{R}$, thus the infinite dimensional basis $J:=\{1, x, x^2,...\}$ is a subset of $H$. The basis for $P_3$ is $K:=\{1,x,x^2,x^3\}$, with dimension 4. Thus $K\subset H$, which implies that $P_3\subset C[0,1]$.