TeXipedia

sup

Denotes the supremum (least upper bound) of a set or function, representing the smallest value that is greater than or equal to all elements.

Overview

Essential in mathematical analysis and optimization theory for finding maximum bounds of sets that may not have a direct maximum value.

  • Commonly used in real analysis to describe properties of functions and sets
  • Critical in optimization problems where exact maximums might not exist
  • Appears frequently in functional analysis and measure theory
  • Often paired with infimum (inf) for describing bounds of sets
  • Particularly useful when dealing with infinite sets or continuous functions

Examples

Finding the supremum of a sequence approaching 1.

sup⁡n→∞(1−1n)=1\sup_{n \to \infty} \left(1 - \frac{1}{n}\right) = 1 
\sup_{n \to \infty} \left(1 - \frac{1}{n}\right) = 1

Supremum of a continuous function on a closed interval.

sup⁡x∈[a,b]f(x)≤M\sup_{x \in [a,b]} f(x) \leq M 
\sup_{x \in [a,b]} f(x) \leq M

Essential supremum in functional analysis.

∥f∥∞=sup⁡x∈X∣f(x)∣\|f\|_{\infty} = \sup_{x \in X} |f(x)| 
\|f\|_{\infty} = \sup_{x \in X} |f(x)|