Let $V$ be a vector space and $M$ be subspace of it. If $f$ is a linear functional on $M$, is it possible to extend it to the whole space $V$? If we have a sublinear functional $p$ on $V$ dominating $f$ on $M$, then by Hahn Banach we know that there is an extension. I would like to know if the same statement is valid with out the hypothesis of domination sublinear functional. I would like to understand the role of sublinear functional in the proof of Hahn Banach. Thanks
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The primary purpose is to ensure that when a space has a norm, we can extend a bounded functional to another bounded linear functional. Much of functional analysis revolves around bounded linear functionals.
As pointed out in comments, if we do not care about boundedness/majorization, the result is also true, and easier to prove. Also, not as useful.