# Equations with Finite Defect

• S. G. Krein

## Abstract

Let F1 be a subspace of the Banach space F. The dimension of the orthogonal complement F 1 is called the defect or the codimension of the subspace F1:
$$def{F_1} = \dim F_1^ \bot$$
This number can be finite or infinite. On the assumption that def F1 = n < ∞, let g1,...,gn be a basis for F 1 If u1F1, then one can construct a linear functional Φ1, ∈ F 1 * such that Φ1(u1) = 1 and Φ1(y) = 0 for all y ∈ F1. Then Φ1 ∈ F 1 . Now let u2 be an element which is not in the linear span of the subspace F1 together with the element u1 (which is again a subspace of F), and choose Φ2∈(F 1 such that Φ2(u2) = 1, Φ2(u1) = 0. This process may be continued, and clearly it will have at most n steps: the functionals Φ1. are linearly independent (if $$\sum {{c_i}\phi (z) = 0}$$ for all z ∈ F, then taking successively z = u1, z = u2,..., we obtain c1 = 0, c2 = 0,...) and are all in F 1 . In fact, the number of steps is exactly n. Indeed, if the process terminates at the m-th step, then one can write every element z ∈ F as
$$z = \sum\nolimits_k^m {1{}^\alpha k} {}^uk + y\;\;(y \in {F_1})$$
.

## Keywords

Banach Space Orthogonal Complement Linear Span Linear Manifold Null Solution