# Definition:Little-Omega Notation

## Definition

### Informal 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.

### Definition 1

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:

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

### Definition 2

Let $f: \N \to \R, g: \N \to \R$ be two real sequences, expressed here as real-valued functions on the set of natural numbers $\N$.

Let:

$\ds \lim_{n \mathop \to \infty} {\frac {\map f n} {\map g n} } = \infty$

Then:

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

### Definition 3

Let $f: \N \to \R, g: \N \to \R$ be two real sequences, expressed here as real-valued functions on the set of natural numbers $\N$.

Then:

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

where $\oo$ denotes little-$\oo$ notation.

A function $f$ is $\map \omega g$ if and only if $f$ is not $\map \OO g$ where $\OO$ is the big-$\OO$ notation.

## 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:

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

## 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.