Extended real number system and conventions

Conventions for inf/sup and extended-real-valued functions used in convex analysis

Extended real number system and conventions

The extended real number system is

\overline{\mathbb{R}}:=\mathbb{R}\cup\{-\infty,\infty\}.

In the notes, for convex analysis it is convenient to work mostly with the one-sided extension

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

so that expressions like $-\infty+\infty$ never arise.

Infimum/supremum conventions (in $\overline{\mathbb{R}}$ ).

$-\infty$ is a lower bound of every subset; every nonempty set has a greatest lower bound.
If a nonempty set is not bounded below, its infimum is $-\infty$ .
By convention, $\inf\emptyset=\infty$ .
$\infty$ is an upper bound of every subset; every nonempty set has a least upper bound.
If a nonempty set is not bounded above, its supremum is $\infty$ .
By convention, $\sup\emptyset=-\infty$ .

Context. Allowing the value $\infty$ lets one encode constraints by penalties (e.g., the indicator function ) and avoid repeatedly restricting domains by hand.