Let $f : X \rightarrow R$ be a mapping on a real vector space X.
Suppose that $f$ is sublinear but not linear, i.e., there exists $x^{*} \in X$ such that $f(−x^{*}) \neq -f(x^{*})$. Use the Hahn-Banach Theorem to show that there exist two different linear forms $L_{1}$ and $L_{2}$ on X such that $L_{1}(x) \leq f(x)$ and $L_{2}(x) \leq f(x)$ for all x ∈ X.
I am trying to prove this statement. So, far what I have done is that I have defined 2 subspaces on X.
$ W=\{x^{*}| f(−x^{*}) \neq -f(x^{*})\} $ & $V$ be the trivial {0} subspace.
Let $l_{1}$ be a linear mapping on $W$ and $l_{2}$ be a linear mapping on V. Then by Hahn Banach Theorem there exists linear map, $L_{1}$ & $L_{2}$ on X such that $L_{1}(x) \leq f(x)$ and $L_{2}(x) \leq f(x)$ for all x ∈ X.
But I don't understand how to show $L_{1}$ & $L_{2}$ are different/distinct maps. Any help will be highly appreciated and thanks in Advance. Please also feel free to comment on what I have done so far.