= Solution
With $K\ge2$ distinct knots $\kappa_1<\cdots<\kappa_K$, a <natural cubic spline> is a $C^2$ function on the real line, cubic on each interval between consecutive knots and affine outside the two outer knots. The latter condition gives $m''(\kappa_1)=m''(\kappa_K)=0$.
There are $4(K-1)$ coefficients for the interior cubic pieces. Matching values and the first two <derivatives> at the $K-2$ interior knots imposes $3(K-2)$ independent constraints, and the two natural endpoint conditions impose two more. The affine continuations are then determined by endpoint values and slopes. The <dimension of a vector space> is therefore
$$
\boxed{4(K-1)-3(K-2)-2=K.}
$$
One explicit <basis> uses $1,x$ and $d_j(x)-d_{K-1}(x)$ for $j=1,\ldots,K-2$, where
$$
d_j(x)=\frac{(x-\kappa_j)_+^3-(x-\kappa_K)_+^3}{\kappa_K-\kappa_j},\qquad u_+=\max(u,0).
$$
The cubic and quadratic terms cancel in each difference above $\kappa_K$, making it affine there; below $\kappa_1$ the truncated powers vanish. The changes in third derivative at individual knots show <independence> of these basis functions. This count uses the <linear independence> of the spline constraints and counts the listed outer knots as boundary knots. If a convention instead lists $K$ interior knots and two additional boundaries, the dimension is $K+2$. With only one listed knot and linearity on both sides, the $C^2$ condition forces a globally <affine function>, so the exceptional dimension is two.
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