Table of contents |

2 Metric spaces 3 Functional analysis |

A set *S* of real numbers is called **bounded above** if there is a real number *k* such that *k* >* s* for all *s* in *S*. The number *k* is called an **upper bound** of *S*. The terms **bounded below** and **lower bound** are similarly defined. A set *S* is **bounded** if it is bounded both above and below. Therefore, a set is bounded if it is contained in a finite interval.

A function *f* : *X* -> **R** is bounded on *X* if its image *f*(*X*) is a bounded subset of **R**.

A set *S* in a metric space (*M*, *d*) is **bounded** if it is contained in a ball of finite radius, i.e. if there exists *x* in *M* and *r* > 0 such that for all *s* in *S*, we have d(*x*, *s*) < *r*.

A set *S* in a topological vector space is **bounded** if it is contained in some multiple of every basic neighbourhood of zero. A bounded linear operator is continuous.