bounded (original) (raw)

Let X be a subset of ℝ. We say that X is bounded when there exists a real number M such that |x|<M for all x∈X. When X is an interval, we speak of a bounded interval.

This can be generalized first to ℝn. We say that X⊆ℝn is bounded if there is a real number M such that ∥x∥<M for all x∈X and ∥⋅∥ is the Euclidean distance between x and y.

This condition is equivalent to the statement: There is a real number T such that ∥x-y∥<T for all x,y∈X.

A further generalization to any metric space V says that X⊆V is bounded when there is a real number M such that d⁢(x,y)<M for all x,y∈X, where d is the metric on V.