Definition:Little-O Notation/Sequence/Definition 1

Definition

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

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

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

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

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

Similarly, when expressed in the notation of sequences, $a_n \in \map \oo {b_n}$ is read as:

$a_n$ is little-$\oo$ of $b_n$

While it is correct and accurate to write:

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

or:

$a_n \in \map \oo {b_n}$

it is a common abuse of notation to write:

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

or:

$a_n = \map \oo {b_n}$

This notation offers some advantages.

The little-$\oo$ notation, along with big-$\OO$ notation, are also referred to as Landau's symbols or the Landau symbols, for Edmund Georg Hermann Landau.