A Dominated Linear Functional on a Line

In the setting of Separable Real Normed Spaces and Hahn–Banach: Notation, let $X$ be a real vector space with zero vector $0_{X}$, let $p$ be a sublinear functional on $X$, let $x_{0}\in X$ with $x_{0}\neq0_{X}$, and let $c$ be a real number with

$$-p(-x_{0})\le c\le p(x_{0}).$$

Let $\mathbb{R}x_{0}=\{t\,x_{0}:t\in\mathbb{R}\}$.

(i) The line. $\mathbb{R}x_{0}$ is a linear subspace of $X$, and if $t,t'\in\mathbb{R}$ satisfy $t\,x_{0}=t'\,x_{0}$, then $t=t'$.

(ii) The functional. The function $g:\mathbb{R}x_{0}\to\mathbb{R}$ given by

$$g(t\,x_{0})=t\,c\qquad(t\in\mathbb{R})$$

is well defined by (i), it is a linear functional on $\mathbb{R}x_{0}$, and $g(y)\le p(y)$ for every $y\in\mathbb{R}x_{0}$.