= Solution
{c}
A <cubic spline> is a twice continuously differentiable function whose restriction to each knot interval is a <polynomial> of degree at most three. Its second derivative is continuous and piecewise linear. Natural endpoint conditions are $C''(x_0)=C''(x_n)=0$.
Linearity of both the piecewise-polynomial and smoothness conditions, and of the endpoint equations, makes the natural splines a <vector space> $V$. A continuous piecewise linear second derivative with zero endpoint values is determined freely by its $n-1$ interior nodal values. Integrating twice adds two arbitrary affine constants. Hence
$$
\boxed{\dim V=(n-1)+2=n+1.}
$$
This count includes the case of one interval.
An interpolating <spline> has $S(x_i)=f_i$ at every node. Set $e=T-S$, so $e$ vanishes at all nodes. On each open knot interval $S^{(3)}$ is constant. Two integrations by parts give
$$
\int_{x_{i-1}}^{x_i}e''S''dx
=[e'S''-eS^{(3)}]_{x_{i-1}}^{x_i}.
$$
The terms containing $e$ vanish at nodes. When summed, the interior $e'S''$ terms cancel because $e'$ and $S''$ are continuous. The two outer terms vanish by the natural endpoint conditions. Thus
$$
\boxed{\int_{x_0}^{x_n}(T''-S'')S''dx=0.}
$$
Integrating the given square identity now yields the sharper energy decomposition
$$
\int_{x_0}^{x_n}(T'')^2dx
=\int_{x_0}^{x_n}(S'')^2dx+\int_{x_0}^{x_n}(T''-S'')^2dx.
$$
In particular \b[the natural interpolating <spline> minimizes the squared-curvature integral].
If $T$ is another natural interpolant, the same minimizing argument in both directions makes the two energies equal. The final nonnegative integral then vanishes, so $T''=S''$ everywhere by <continuity>. Their difference is affine and zero at all knots, including the two distinct endpoints, so $T=S$. This proves uniqueness.
It also supplies existence if needed: nodal evaluation is a <linear map> from the $(n+1)$-dimensional space $V$ to $\mathbb R^{n+1}$. Its <kernel> is zero by the same argument with zero data and the zero competitor. It is therefore <surjective>, so every data vector has a natural <spline> interpolant.
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