Definition:Little-O Notation/Sequence
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 $\N$.
$\oo$-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.
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 $\map \oo g$ is defined as:
- $\map \oo g = \set {f: \N \to \R: \forall c \in \R_{>0}: \exists n_0 \in \N: \forall n > n_0: \size {\map f n} \le c \cdot \size {\map g n} }$
Definition 2
Let $\sequence {a_n}$ and $\sequence {b_n}$ be sequences of real or complex numbers.
Let $b_n \ne 0$ for all $n$.
$a_n$ is little-$\oo$ of $b_n$ if and only if:
- $\ds \lim_{n \mathop \to \infty} \frac {a_n} {b_n} = 0$
Notation
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.
Also known as
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.