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Could you please give me hints may leads to prove the following:

Let $X$ be a real vector space, $\,p_1,p_2:X\to\mathbb R\,$ be two sublinear functionals, and $\,f:X\to\mathbb R\,$ be a linear functional satisfying $$ f(x)\le p_1(x)+p_2(x), \quad\text{for all $\,x\in X$.} $$ Prove that there exist two linear functionals $\,f_1,f_2:X\to\mathbb R$, such that $\,f=f_1+f_2\,$ and $\,f_i(x)\le p_i(x),\,$ for $i\in\{1,2\}$ and $x\in X$.

I think we should use Hahn-Banach Theorem.

Thanks in advance.

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marked as duplicate by Jonas Meyer, Rolf Hoyer, user147263, user99914, Claude Leibovici May 4 '15 at 6:22

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