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I find it strange that it is such a useful property for a linear functional to be bounded by a sublinear functional. What information does it really give?

Let $X$ be a Banach space and $p,f:X\to\mathbb{R}$. Well I realize that if $p$ is a sublinear functional and $f$ is a linear one, the condition $|f(x)|\le p(x)$ is weaker than the condition $||f||<\infty$. But what does it really imply? I mean ok, if $p$ is bounded then $f$ is bounded, but what is $p$ is not bounded?

For example, if $f$ is defined on $V$ a subspace of $X$ and we take $\lim_{\,n\to\infty}f(x_n)$, with $x_n\to x\in\overline{V}\backslash V$ can we deduct that the limit exists (with the possibility of being infinity) or it may not even exist at all? Do we get any information on the spectrum of $f$?

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1 Answer 1

I think that the main point of the condition is that Hahn Banach applies in more generality than normed spaces.

For example one can prove that any two convex sets in a topological vector space (not even locally convex), with one of them open, can be separated by a continuous functional (i.e "geometric Hahn-Banach theorem"). The proof uses Hahn Banach where $p$ is the Minkowski functional of a well-chosen neighbourhood of zero.

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