Sometimes we have to deal with equations (A) such that one can see from their structure immediately that they cannot be solved for all y ∈ F. For example, this happens if there is a closed operator Φ, acting from F into another Banach space G, and such that ΦA = 0. Then equation Ax = y can be solved only for y ∈ N(Φ). Relative to a given pair of spaces E and F, an equation which can be solved only when the right-hand side is contained in a subspace F, of F, must be naturally considered as overdetermined.
KeywordsBanach Space Homogeneous Equation Orthogonal Complement Closed Operator Unique Solvability
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