Definition:Little-O Notation/Sequence/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 $\N$.

$\oo$-notation is used to define an upper 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:

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

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

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.

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