Three problems arising in approximation theory are studied. These problems have already been studied by Arthur Sard. The main goal of this paper is to use geometrical compatibility theory to extend Sard's results and get characterizations of the sets of solutions.

We study the problem of approximation by the sets S + K(H), $S_e$, V + K(H) and $V_e$ where H is a separable complex Hilbert space, K(H) is the ideal of compact operators, $S=L\in B\left(H\right):L*L=I$ is the set of isometries, V = S ∪ S* is the set of maximal partial isometries, $S_e=L\in B\left(H\right):\pi (L*)\pi \left(L\right)=\pi \left(I\right)$ and $V_e=S_e\cup S_e*$ where π : B(H) → B(H)/K(H) denotes the canonical projection. We also prove that all the relevant distances are attained. This implies that all these classes are closed and we remark that $V_e=V+K\left(H\right)$. We also show that S + K(H) is both closed and open in $S_e$....