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The dual space isthe vector space of linear functionals on $V$. If $e_1,\dots ,e_n$ is a basis of a finite-dimensional $V$, define $e_i^{\ast }(e_j)={\delta }_{ij}$. Every $f\in V^{\ast }$ has the unique expansionso the dual basis $e_1^{\ast },\dots ,e_n^{\ast }$ proves

For $U\le V$, its annihilator of a vector subspace isIf $u_1,\dots ,u_k$ is a basis of $U$ and is extended to a basis $u_1,\dots ,u_n$ of $V$, thenConsequentlyIf $U\ne V$, this dimension is positive, giving a nonzero functional that vanishes on $U$.

For a linear map $\alpha :V\to W$, the dual map isNowsoEvery $g\circ \alpha$ vanishes on $\ker \alpha$, henceThe two spaces have the same dimension, sinceThus

Let $Q:V\to V/U$ be the quotient map. Since $Q$ is onto, $Q^{\ast }$ is injective, and the preceding identity givesThereforeFor the inclusion $J:U\hookrightarrow V$, the map $J^{\ast }:V^{\ast }\to U^{\ast }$ is restriction to $U$. It is onto and has kernelThe first isomorphism theorem now gives the other duals of a subspace and its quotient:

Fix ${\xi }_i\in [t_i,t_{i+k}]$ and choose a knot interval $I=(t_{\ell },t_{\ell +1})$ adjacent to it on which $N_i$ is active. Exactly $k$ B-splines are nonzero on $I$, and their polynomial restrictions form a basis of ${\mathcal{P}}_{k-1}$. Applying the expansion from part (a) to the polynomial piece $N_j{|}_I$ givesUniqueness of coordinates in this local basis forceswhen $N_j$ is active. If $N_j$ vanishes on $I$, all of its local polynomial derivatives vanish and the same equality holds with value zero. At a knot, use either adjacent polynomial piece; the $x$-independence proved in part (a) gives the same coefficient. Henceso the ${\lambda }_i$ form the dual basis to the B-spline basis.

For a basis $(u_i)$ of a finite-dimensional inner-product space with conjugate-linear first argument, define $G_{ij}=(u_i,u_j)$ and ${\hat{u}}_k={\sum }_l(G^{-1}{)}_{lk}{u}_l$. Then $(u_i,{\hat{u}}_k)={\delta }_{ik}$ and $({\hat{u}}_j,{\hat{u}}_k)=(G^{-1}{)}_{jk}$. This is the inner-product representation of the dual basis; requiring the dual vectors to lie in the given span is essential.