The Hahn–Banach Theorem in a Separable Real Normed Space

In the setting of Separable Real Normed Spaces and Hahn–Banach: Notation, let $X$ be a separable real normed space, let $p$ be a sublinear functional on $X$, and let $C\ge0$ be a real number with

$$p(x)\le C\,\lVert x\rVert\qquad\text{for every }x\in X.$$

Let $M$ be a linear subspace of $X$ and let $f$ be a linear functional on $M$ with $f(x)\le p(x)$ for every $x\in M$.

(i) Dominated extension. There is an element $F$ of the dual space $X^{*}$ with

$$F(x)=f(x)\quad\text{for every }x\in M,\qquad F(x)\le p(x)\quad\text{for every }x\in X.$$