# A question about sublinear functionals [duplicate]

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 $x\in X$.