A basis for space of spline of degree k how can i proof space of spline of degree k on $[x_0,x_n]$ with knots:
$$x_0<x_1<\cdots<x_{n-1}<x_n,$$
have a basis like this:
$$\{1,x,x^2,\ldots,x^k,(x-x_1)^k_+,(x-x_2)^k_+,\ldots,(x-x_{n-1})^k_+\}.$$
remark:
$$(x-x_i)^k_+=\left\{\begin{array}{2}(x-x_i)^k& x>x_i\\0& x\leq x_i\end{array}\right.$$

Kincaid D., Cheney W. Numerical analysis, Chapter 6:
A natural spline of degree $2m+1$ is a function $s\in C^{2m}(\mathbb{R})$
that reduces to a polynomial of degree $\leq2m+1$ in each inner interval
and to a polynomial of degree at most $m$ in $(-\infty,t_{1})$ and
$(t_{n},\infty)$.
Theorem. Every natural spline can be represented using truncated power functions
as follows
$$
s(x)=\sum_{k=0}^{m}\alpha_{k}x^{k}+\sum_{i=1}^{n}\beta_{i}(x-x_{i})_{+}^{2m+1}
$$
with the coefficient conditions,
$$
\sum_{i=1}^{n}\beta_{i}x_{i}^{k}=0,\qquad k=0,1,\ldots m
$$
