Extended real number system and conventions

The extended real number system adjoins two formal endpoints to the real numbers:

\overline{\mathbb R}=\mathbb R\cup\{-\infty,+\infty\},

ordered by $-\infty<x<+\infty$ for every $x\in\mathbb R$ . Convex analysis often uses the one-sided extension

\mathbb R\cup\{+\infty\}=(-\infty,+\infty]

for functions that encode infeasible points by the value $+\infty$ .

Infimum and supremum conventions

Every nonempty $A\subseteq\overline{\mathbb R}$ has a greatest lower bound and a least upper bound in the extended order, extending the real infimum and supremum. If $A$ has no real lower bound then $\inf A=-\infty$ ; if it has no real upper bound then $\sup A=+\infty$ . The distinction between real and extended bounds matters: $-\infty$ is an extended lower bound for every subset. For the empty set, the conventions are

\inf\varnothing=+\infty, \qquad \sup\varnothing=-\infty.

Remarks

Allowing the value $+\infty$ lets an indicator function encode a constraint without repeatedly restricting the domain.