One-Step Extension of a Dominated Linear Functional

In the setting of Separable Real Normed Spaces and Hahn–Banach: Notation, let $V$ be a real vector space, let $p$ be a sublinear functional on $V$, let $M$ be a linear subspace of $V$, let $x_{0}\in V\setminus M$, and let $f$ be a linear functional on $M$ with $f(m)\le p(m)$ for every $m\in M$. Let

$$M+\mathbb{R}x_{0}=\{m+t\,x_{0}:m\in M,\ t\in\mathbb{R}\}.$$

(i) The enlarged subspace. $M+\mathbb{R}x_{0}$ is a linear subspace of $V$ containing $M$ and $x_{0}$.

(ii) Unique representation. If $m,m'\in M$ and $t,t'\in\mathbb{R}$ satisfy $m+t\,x_{0}=m'+t'\,x_{0}$, then $m=m'$ and $t=t'$.

(iii) The extension constant. The set $\{f(m)-p(m-x_{0}):m\in M\}$ is nonempty and bounded above, and its supremum $c$ satisfies

$$f(m)-p(m-x_{0})\le c\le p(m+x_{0})-f(m)\qquad\text{for every }m\in M.$$

(iv) The extension. With $c$ as in (iii), the function $g:M+\mathbb{R}x_{0}\to\mathbb{R}$ given by

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

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