Theorem: Finite matching and infeasibility certificates

For the fixed package of Definition def:finite:

1. If $r(D)\ne 0$, the selected pair fails exact matching. This does not imply that every other $a\in X$ fails.

2. A matching element exists in $X$ if and only if

   $${\kappa }_{\mathrm{cmp}}s(D)\in \mathrm{range}L.$$

3. Infeasibility is equivalent to existence of $\ell \in Y^{\ast }$ such that

   $$\ell \circ L=0,\ell ({\kappa }_{\mathrm{cmp}}s(D))\ne 0.$$

4. With an auxiliary Euclidean inner product on $Y$,

   $$\underset{a\in X}{\inf }\Vert La-{\kappa }_{\mathrm{cmp}}s(D)\Vert =\Vert {\Pi }_{(\mathrm{range}L{)}^{\perp }}({\kappa }_{\mathrm{cmp}}s(D))\Vert .$$

   The infimum is attained.

Proof. The first assertion is the definition of the selected residual. The second is the definition of the image of $L$.

If the source vector is outside the finite-dimensional subspace $\mathrm{range}L$, project it onto the orthogonal complement of that subspace and pair with this nonzero projection. This gives (eq:left-witness). Conversely, such a functional excludes membership in the image.

Orthogonal decomposition of the source vector shows that its component perpendicular to $\mathrm{range}L$ is unavoidable in every residual. Its parallel component belongs to the image and has a preimage in $X$, proving (eq:distance) and attainment.