Definition:O Notation/Little-O Notation

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Definition

Little-O notation occurs in a variety of contexts.

Sequences

Let $\sequence {a_n}$ and $\sequence {b_n}$ be sequences of real or complex numbers.

Let $b_n\neq0$ for all $n$.


$a_n$ is little-O of $b_n$ if and only if

$\ds \lim_{n \mathop \to \infty} \frac {a_n} {b_n} = 0$


Real Functions

Let $f$ and $g$ be real-valued or complex-valued functions on a subset of $\R$ containing all sufficiently large real numbers.

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


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

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


Point Estimate

Definition:O Notation/Little-O Notation/Real Point

Complex Functions

Definition:O Notation/Little-O Notation/Complex Functions

Complex Point Estimate

Definition:O Notation/Little-O Notation/Complex Point

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

$\map f x = \map o {\map g x}$ 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 $x \in U$


Sources