I have been seeing various versions of Hahn Banach theorem . There are few confusions that i would like to clarify . 1) In some proof they assume the existence of a sublinear functional and the functional $f$ must satisfy $\le$ condition to the sublinear functional $p$ . Does there always exist a sublinear functional ? In some proofs they don't say anything about the sublinear functional , is it wrong ?
2) I am not able to appreciate the induction , because it assumes the existence of the extension of linear functional and finally says that maximal element should be the Space itself ? I particularly have a problem in applying the first induction step.
I would like to have some knowledge about my confusions and misunderstandings . Thanks