Equations with Finite Defect

  • S. G. Krein


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


Banach Space Orthogonal Complement Linear Span Linear Manifold Null Solution 


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Copyright information

© Birkhäuser Boston, Inc. 1982

Authors and Affiliations

  • S. G. Krein
    • 1
  1. 1.Department of MathematicsVoronezh UniversityVoronezhUSSR

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