Appendix: Functional and Real Analysis Tools
Antonino Maugeri, Dian K. Palagachev, Lubomira G. Softova · 2000
We have collected here, for the sake of readers' convenience, some fundamental results used frequently in the exposition above.The corresponding proofs can be found at the monographs cited. A.I. Functional and Real Analysis ToolsTheorem A.I. (Method of Continuity) ([GT, Theorem 5.2]) Let B be a Banach space, Af a normed linear space and let I/o, LI be bounded linear operators from B into Af.For each t G [0,1], set L t = (l-t)L Q + tLi and suppose that there is a constant C such that \\x\\ B < C\\L t x\\tf for t G [0,1].Then LI maps B onto M if and only if LQ maps B onto JV.A continuous mapping between two Banach spaces is called compact (or completely continuous) if it maps bounded sets into precompact (that is, their closures are compact).