Three non-linear problems on normed spaces.

摘要

In this dissertation, we will study the following three non-linear problems: (1) The lineability problem for functionals. (2) The minimum-norm problem for translations. (3) The Banach-Mazur conjecture for rotations.; As far as we know, all of them are currently open, and we believe that any approach to their solutions will constitute a work of great interest to the mathematical community. In this dissertation, we obtain progresses that lead to partial solutions of these problems.