Could you please give me hints may leads to solve the following question:
Let \(\displaystyle X_{| \mathbb{R}}\) be a real vector space, \(\displaystyle p_1,\, p_2\, :\, X\, \rightarrow\, \mathbb{R}\) be two sublinear functionals, and \(\displaystyle f\, :\, X\, \rightarrow\, \mathbb{R}\) be a linear functional satisfying:
. . . . .\(\displaystyle f(x)\, \leq\, p_1(x)\, +\, p_2(x),\, \forall\, x\, \in\, X\)
Prove that there exist two linear functionals \(\displaystyle f_1,\, f_2\, :\, X\, \rightarrow\, \mathbb{R}\) such that \(\displaystyle f\, =\, f_1\, +\, f_2\) and \(\displaystyle f_i(x)\, \leq\, p_i(x),\) for all \(\displaystyle i\, \in\, \{1,\, 2\}\) and \(\displaystyle x\, \in\, X.\)
(I think we should use Hanan Banach theorem).
Thanks in advance.
Let \(\displaystyle X_{| \mathbb{R}}\) be a real vector space, \(\displaystyle p_1,\, p_2\, :\, X\, \rightarrow\, \mathbb{R}\) be two sublinear functionals, and \(\displaystyle f\, :\, X\, \rightarrow\, \mathbb{R}\) be a linear functional satisfying:
. . . . .\(\displaystyle f(x)\, \leq\, p_1(x)\, +\, p_2(x),\, \forall\, x\, \in\, X\)
Prove that there exist two linear functionals \(\displaystyle f_1,\, f_2\, :\, X\, \rightarrow\, \mathbb{R}\) such that \(\displaystyle f\, =\, f_1\, +\, f_2\) and \(\displaystyle f_i(x)\, \leq\, p_i(x),\) for all \(\displaystyle i\, \in\, \{1,\, 2\}\) and \(\displaystyle x\, \in\, X.\)
(I think we should use Hanan Banach theorem).
Thanks in advance.
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