(abbr. Inf). The greatest lower bound on a (real-valued) function over (a subset of) its domain. If f is unbounded from below, Inf{f} = - infinity, and if the domain is empty, Inf{f} = infinity. Otherwise, suppose L is any lower bound: f(x) = L for all x in X. L is a greatest lower bound if, for any e 0, there exists x in the domain for which f(x) = L+e. (That is, we can get arbitrarily close to L in the range of f.) Note that the infimum is always defined, and its range is in the extended reals. It is the minimum, if it is attained by some point in its domain.

For a poset and a subset of , the greatest element in the set of lower bounds of (if it exists, which it may not) is called the infimum, meet, or greatest lower bound of . It is denoted by inf or . The infimum of two elements may be written as inf{,} or ∧ . If the set is finite, one speaks of a finite infimum. The dual notion is called supremum.

In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is less than or equal to all other elements of the subset. Consequently the term greatest lower bound (also abbreviated as glb or GLB) is also commonly used. Infima of real numbers are a common special case that is especially important in analysis.