Theorem

The set of spline functions of degree l on $\sigma_{n}^{}$ is a $\mathbb {R}$-vectorial subspace of dimension n + l.

Proof
On [a, x₁], s is a polynomial A of degree less or equal to l, hence on [a, x₁], s = A(x) = a₀ + a₁x + ...a_lx^l and A is a linear combinaison of 1, x,...x^l.
On [x₁, x₂], s is a polynomial B of degree less or equal to l, hence on [x₁, x₂], s = B(x) = b₀ + b₁x + ...b_lx^l.
s has continuous derivatives up to order l - 1, hence :

$$\forall$$0 $$\leq$$ j $$\leq$$ l - 1,    B^(j)(x₁) - A^(j)(x₁) = 0

therefore B(x) - A(x) = $\alpha_{1}^{}$(x - x₁)^l or B(x) = A(x) + $\alpha_{1}^{}$(x - x₁)^l.
Define the function :

q₁(x) = $$\left\{\vphantom{ \begin{array}{rcl} 0 & \mbox{sur} & [a,x_1] \\ (x-x_1)^l & \mbox{sur} & [x_1,b]\\ \end{array} }\right.$$$$\begin{array}{rcl} 0 & \mbox{sur} & [a,x_1] \\ (x-x_1)^l & \mbox{sur} & [x_1,b]\\ \end{array}$$

Hence :

s|_{[a, x2]} = a₀ + a₁x + ...a_lx^l + $$\alpha_{1}^{}$$q₁(x)

On [x₂, x₃], s is a polynomial C of degree less or equal than l, hence on [x₂, x₃], s = C(x) = c₀ + c₁x + ...c_lx^l.
s has continuous derivatives until l - 1, hence :

$$\forall$$0 $$\leq$$ j $$\leq$$ l - 1,    C^(j)(x₂) - B^(j)(x₂) = 0

therefore C(x) - B(x) = $\alpha_{2}^{}$(x - x₂)^l or C(x) = B(x) + $\alpha_{2}^{}$(x - x₂)^l.
Define the function :

q₂(x) = $$\left\{\vphantom{ \begin{array}{rcl} 0 & \mbox{on} & [a,x_2] \\ (x-x_2)^l & \mbox{on} & [x_2,b]\\ \end{array} }\right.$$$$\begin{array}{rcl} 0 & \mbox{on} & [a,x_2] \\ (x-x_2)^l & \mbox{on} & [x_2,b]\\ \end{array}$$

Hence : s|_{[a, x3]} = a₀ + a₁x + ...a_lx^l + $\alpha_{1}^{}$q₁(x) + $\alpha_{2}^{}$q₂(x)
And so on, the functions are defined by :

$$\forall$$1 $$\leq$$ j $$\leq$$ n - 1,q_j(x) = $$\left\{\vphantom{ \begin{array}{rcl} 0 & \mbox{on} & [a,x_j] \\ (x-x_j)^l & \mbox{on} & [x_j,b]\\ \end{array} }\right.$$$$\begin{array}{rcl} 0 & \mbox{on} & [a,x_j] \\ (x-x_j)^l & \mbox{on} & [x_j,b]\\ \end{array}$$

hence,

s|_{[a, b]} = a₀ + a₁x + ...a_lx^l + $$\alpha_{1}^{}$$q₁(x) + .... + $$\alpha_{{n-1}}^{}$$q_n-1(x)

and s is a linear combination of n + l independant functions 1, x,..x^l, q₁,..q_n-1.