Definition:Little-O Notation

Definition

Little-$\oo$ notation occurs in a variety of contexts.

Sequences

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:

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

Estimate at Infinity

Let $f$ and $g$ be real functions defined on a neighborhood of $+\infty$ in $\R$.

Let $\map g x \ne 0$ for $x$ sufficiently large.

$f$ is little-$\oo$ of $g$ as $x \to \infty$ if and only if:

$\ds \lim_{x \mathop \to \infty} \frac {\map f x} {\map g x} = 0$

Complex Point Estimate

Let $f$ and $g$ be complex functions.

Let $z_0 \in \C$ be such that there exists a punctured neighborhood of $z_0$ on which $f$ and $g$ are defined.

We say $f$ is little-$\oo$ of $g$, and denote:

$\map f z \in \map \oo {\map g z} \text{ as } z \to z_0$

if and only if:

$\ds \lim_{z \mathop \to z_0} \dfrac {\size {\map f z} } {\size {\map g z} } = 0$

where $\size {\, \cdot \,}$ denotes the complex modulus.

General Definition for point estimates

Let $X$ be a topological space.

Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\norm {\,\cdot\,}$.

Let $f, g: X \to V$ be functions.

Let $x_0 \in X$.

The statement

$\map f x = \map \oo {\map g x}$ as $x \to x_0$

is equivalent to the statement:

For all $\epsilon > 0$, there exists a neighborhood $U$ of $x_0$ such that $\norm {\map f x} \le \epsilon \cdot \norm {\map g x}$ for all $x \in U$

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.

Also see

• Definition:Big-$\OO$ Notation

• Definition:Big-Omega Notation
• Definition:Little-Omega Notation

• Definition:Theta Notation

• Results about little-$\oo$ notation can be found here.

Sources

• 1994: H.E. Rose: A Course in Number Theory (2nd ed.) ... (previous) ... (next): Preface to first edition: Prerequisites