Let $\Delta$ be a subdivision of $[a,b]$:
$a=t_1 < t_2 <\ldots < t_N =b$. Then
$\mathbb{S}_{m}^{k}(\Delta )=\{s\in \mathcal{C}^{k}([a,b]) : s_{\mid_{[t_i , t_{i+1}]}}\in\mathbb{P}_{m}, i=1,\ldots ,N-1\}$
where $\mathbb{P}_{m}$ is the linear space of Polynomials of degree $≤m$.
Prove that $\mathbb{S}_{m}^{m}(\Delta )=\mathbb{P}_m$
Obviously, $\mathbb{P}_m\subseteq\mathbb{S}_{m}^{m}(\Delta )$ but in the other case I have no idea. The problem in the splines are the points $t_i$ but in this case we don't have any problems of differentiability but I don't know what let me see that $s$ is a polynomial. Can someone give me an advice? Thanks before!