Definition:O Notation/Big-O Notation/General Definition

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Definition

Estimate at infinity

Let $\struct {X, \tau}$ 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.


The statement:

$\map f x = \map \OO {\map g x}$ as $x \to \infty$

is equivalent to:

There exists a neighborhood of infinity $U \subset X$ such that:
$\exists c \in {\R}_{\ge 0}: \forall x \in U: \norm {\map f x} \le c \norm {\map g x}$


That is:

$\norm {\map f x} \le c \norm {\map g x}$

for all $x$ in a neighborhood of infinity.


Point Estimate

Let $\struct {X, \tau}$ be a topological space.

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

Let $x_0 \in X$.

Let $f, g: X \setminus \set {x_0} \to V$ be functions.


The statement

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

is equivalent to:

$\exists c \in {\R}_{\ge 0}: \exists U \in \tau: x_0 \in U: \forall x \in U \setminus \set {x_0}: \norm {\map f x} \le c \norm {\map g x}$


That is:

$\norm {\map f x} \le c \norm {\map g x}$

for all $x$ in a punctured neighborhood of $x_0$.


Also known as

The big-$\OO$ notation, along with little-$\mathcal o$ notation, are also referred to as Landau's symbols or the Landau symbols, for Edmund Georg Hermann Landau.


In analytic number theory, sometimes Vinogradov's notations $f \ll g$ or $g \gg f$ are used to mean $f = \map \OO g$.

This can often be clearer for estimates leading to typographically complex error terms.


Some sources use an ordinary $O$:

$f = \map O g$