Definition:Little-Omega Notation/Definition 1

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Let $g: \N \to \R$ be a real sequence, expressed here as a real-valued function on the set of natural numbers $\N$.

Then $\map \omega g$ is defined as:

$\map \omega g = \set {f: \N \to \R: \forall c \in \R_{>0}: \exists n_0 \in \N: \forall n > n_0: 0 \le c \cdot \size {\map g n} < \size {\map f n} }$


The expression $\map f n \in \map \omega {\map g n}$ is read as:

$\map f n$ is little-omega of $\map g n$

While it is correct and accurate to write:

$\map f n \in \map \omega {\map g n}$

it is a common abuse of notation to write:

$\map f n = \map \omega {\map g n}$

This notation offers some advantages.

Also known as

Note that in the Greek alphabet, $\omega$ is the minuscule counterpart of the capital letter $\Omega$.

Hence the former is called little-omega and the latter big-omega.

Some sources, therefore, write $\omega$ notation as little-$\omega$ notation, despite the fact that $\omega$'s "little"-ness is intrinsic.

$\mathsf{Pr} \infty \mathsf{fWiki}$ may sometimes adopt this convention if clarity is improved.

Also see

  • Results about little-$\omega$ notation can be found here.