# 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.

## Notation

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**.

## Sources

- 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