Hahn-Banach Choice of constant $c$ and norm of linear functional. I am puzzled for the choice of constant $l(x)=c$ for the Geometric Hahn-Banach Theorem (Hyperplane Separation Theorem).
Can $c$ be any real number, or is it dependent on the context?
Also, I have seen people controlling the norm of the linear functional $l$, e.g. letting $\|l\|=1$. Why is this allowed?
Background:
The version of Hahn-Banach in my text is:
Let $K$ be a nonempty convex subset of a linear space $X$ over the reals; suppose that all points of $K$ are interior. Any point $y$ not in $K$ can be separated from $K$ by a hyperplane $l(x)=c$; that is there is a linear functional $l$, depending on $y$, such that $l(x)<c$ for all $x\in K$, $l(y)=c$.
I know some sources put $c=1$, but is it possible for us to choose our own $c$?
Can we choose the norm of l, $\|l\|$, to be anything we like?
 A: In this version of Hahn-Banach Theorem:

Theorem $\spadesuit$: Let $K$ be a nonempty convex subset of a linear space $X$ over the reals; suppose that all points of $K$ are interior. Any point $y$ not in $K$ can be separated from $K$ by a hyperplane $l(x)=c$; that is there is a linear functional $l$, depending on $y$, such that $l(x)<c$ for all $x\in K$, $l(y)=c$.

Let's assume $l$ as the linear functional of the Theorem $\spadesuit$. If $c=0$, then it is not possible to reformulate Theorem $\spadesuit$ changing $c$ by another constant $C\neq 0$. 
If $c\neq 0$, we know that for all $\alpha \in \mathbb{R}$,  $\alpha l $ is a linear functional, but you should be cautious since not all linear functional of this type $\alpha l $ satisfy the Theorem $\spadesuit$. 
The linear functional  $\alpha l $ satisfying Theorem $\spadesuit$ are those for which $\alpha$ preserve inequality $\alpha l (x)<\alpha  c$. In this sense, for any $C\in \mathbb{R}$  with $C\neq 0$, we can reformulate Theorem $\spadesuit$ changing $c$ by the constant $C$ and $l$ by $L$ where $L=\alpha l$ for appropriate choice of $\alpha$, this choise is:
$$
\alpha=\left\{\begin{array}{rcl}
\frac{C}{c} && \mbox{If } C,c>0 \mbox{ or } C,c<0.  \\
\frac{-C}{c} && \mbox{If } C>0 \mbox{ and } c<0, \mbox{ or }  C<0 \mbox{ and } c>0. \\
\end{array}\right.
$$
