How would you modify the statement of Theorem 7.62 if ker L ≠ {0}?
Theorem 7.62
Suppose L: U → V is a linear map between inner product spaces with ker L = {0} and adjoint map L*: V → U. Let K = L* ○ L: U → U be the associated positive definite operator. If f ∈ rng K, then the quadratic function
p(u) = 1/2 ||L[u]||2 - (u, f) (7.81)
has a unique minimizer u*, which is the solution to the linear system K[u*] = f.