Linear Equations in Banach Spaces pp 17-17 | Cite as

# The Equation Adjoint to the Factored Equation

Chapter

## Abstract

Assume that .

*N*(A) is closed and consider the operator Ã*, the adjoint to the operator Ã from Ẽ =*E*/*N*(A) to F. Then Ã* is defined on υ(Ã*) ⊂ F* and takes its value in Ẽ*. Using the general theorems, Ẽ* =*N*(A)^{⊥}, where the orthogonal complement is taken in E* Let us show that υ(Ã*) and υ(A*) are the same. Indeed, let g ∈ υ(A*) ⊂ F*, i.e., the functional g(Ax) is bounded on υ(A):$$ \left| {g(Ax)} \right| \leqslant c{\left\| x \right\|_E}\quad (x \in D(A)) $$

## Keywords

Banach Space Linear Equation Dimensional Case Homogeneous Equation Orthogonal Complement
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## Copyright information

© Birkhäuser Boston, Inc. 1982