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

## 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 $O$:

- $f = \map O g$