Definition:Supremum/Also known as
Jump to navigation
Jump to search
Definition
Particularly in the field of analysis, the supremum of a set $T$ is often referred to as the least upper bound of $T$ and denoted $\map {\mathrm {lub} } T$ or $\map {\mathrm {l.u.b.} } T$.
Some sources refer to the supremum of a set as the supremum on a set.
Some sources refer to the supremum of a set as the join of the set and use the notation $\bigvee S$.
Some sources introduce the notation $\ds \sup_{y \mathop \in S} y$, which may improve clarity in some circumstances.
Some older sources, applying the concept to a (strictly) increasing real sequence, refer to a supremum as an upper limit.
Sources
- 1919: Horace Lamb: An Elementary Course of Infinitesimal Calculus (3rd ed.) ... (previous) ... (next): Chapter $\text I$. Continuity: $2$. Upper or Lower Limit of a Sequence
- 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Real Numbers: $1.34$. Definition
- 1970: Arne Broman: Introduction to Partial Differential Equations ... (previous) ... (next): Chapter $1$: Fourier Series: $1.1$ Basic Concepts: $1.1.2$ Definitions
- 1982: Peter T. Johnstone: Stone Spaces ... (previous) ... (next): Chapter $\text I$: Preliminaries, Definition $1.2$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): least upper bound