Definition:Little-Omega Notation/Informal Definition

Definition

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

$\omega$-notation is used to define a lower bound for $g$ which is not asymptotically tight.

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

Then:

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

means that $\map f n$ becomes arbitrarily large relative to $\map g n$ as $n$ approaches (positive) infinity.

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.

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.

1990: Thomas H. Cormen, Charles E. Leiserson and Ronald L. Rivest: Introduction to Algorithms ... (previous) ... (next): $2$: Growth of Functions: $2.1$ Asymptotic Notation: $\omega$-notation