# Spline Space question

Why the spline space $$S_{m}^{m}([a,b])=\{s,s \in C^{m}[a,b],s/_{[x_{i},x_{i+1}]}\in P_{m},i=1,..,n\}=P_{m}$$?
I try to take a $$(m-1)^{th}$$ polynomial and i prove that is not in to $$S_{m}^{m}([a,b])$$ but i failed.

Let $$s_{i-1}$$ be the polynomial in the $$[x_{i-1},x_{i}]$$ region of $$[a,b]$$ and $$s_{i}$$ the polynomial in the $$[x_{i},x_{i+1}]$$. Then by the continuity of all the $$m$$ derivatives of $$s$$ in the internal nodes of $$[a,b]$$ we have that:
$$s_{i-1}^{(m)}(x_{i}) = c_{i-1}\quad$$ and$$\quad$$ $$s_{i} ^{(m)}(x_{i}) = c_{i}$$ $$\quad$$ $$\rightarrow$$ $$\quadc_{i-1}= c_{i}$$
$$s_{i-1}^{(m-1)}(x_{i}) =a_{i-1}x_{i}+c_{i}\quad$$ and$$\quad$$ $$s_{i} ^{(m-1)}(x_{i}) = a_{i}x_{i}+c_{i}\quad\rightarrow\quad$$ $$a_{i-1}=a_{i}$$
So we have that $$s_{i-1}=s_{i}$$ i.e. we have that $$s$$ is the same polynomial in each $$[x_{i},x_{i+1}]$$.