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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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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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