Definition:O Notation/Little-O Notation

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Definition

Little-O notation occurs in a variety of contexts.

Sequences

Let $\left \langle {a_n} \right \rangle$ and $\left \langle {b_n} \right \rangle$ 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

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


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 $g(x)\neq0$ for $x$ sufficiently large.


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

$\displaystyle \lim_{x \to \infty} \ \frac{f \left({x}\right)} {g \left({x}\right)} = 0$


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 $\left\Vert{\,\cdot\,}\right\Vert$

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

Let $x_0\in X$.

The statement

$f(x) = o \left({g(x)}\right)$ as $x\to x_0$

is equivalent to the statement:

For all $\epsilon>0$, there exists a neighborhood $U$ of $x_0$ such that $\Vert f(x)\Vert\leq \epsilon\cdot\Vert g(x)\Vert$ for all $x\in U$