TeXipedia

inf

Represents the infimum or greatest lower bound of a set of numbers in mathematical notation.

Overview

Essential in analysis and optimization theory for describing the lowest possible value that bounds a set from below, even when that value is not attainable within the set itself.

  • Commonly used in real analysis to describe bounds of sequences and functions
  • Critical in optimization problems where finding minimum values is important
  • Often paired with supremum (sup) in mathematical proofs and theorems
  • Appears frequently in functional analysis and measure theory

Examples

Finding the infimum of a set of real numbers.

inf⁡{x∈R:x2>2}=−2\inf\{x \in \mathbb{R} : x^2 > 2\} = -\sqrt{2} 
\inf\{x \in \mathbb{R} : x^2 > 2\} = -\sqrt{2}

Expressing the greatest lower bound of a function.

inf⁡x∈[0,1]f(x)=0\inf_{x \in [0,1]} f(x) = 0 
\inf_{x \in [0,1]} f(x) = 0

Defining distance between sets using infimum.

d(A,B)=inf⁡{d(a,b):a∈A,b∈B}d(A,B) = \inf\{d(a,b) : a \in A, b \in B\} 
d(A,B) = \inf\{d(a,b) : a \in A, b \in B\}