Equations with Finite Defect

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,...,g_n be a basis for F ^⊥₁ If u₁ ∉ F₁, 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

$$ z = \sum\nolimits_k^m {1{}^\alpha k} {}^uk + y\;\;(y \in {F_1}) $$

.