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