Let L be a closed operator on a
Hilbert space ℋ defined on a linear manifold 𝒟 of ℋt with the property
that L has a continuous right inverse T and that the dimension of the null
space of L is finite. A boundary functional η for L is defined to be a linear
functional η on 𝒟 such that ηT is continuous. The boundary-value problems for
ordinary differential equations are generalized to the operator L with the
boundary conditions defined by a set of boundary functionals. It is shown, in
particular, that if K is a continuous right inverse of L, then there exist n linearly
independent boundary functionals, η1,⋯,ηn, where n is the dimension of the
null space of L, such that the range of K is precisely the linear manifold
{u𝜖𝒟|ηI(u) = 0,j = 1,2,⋯,n}.
