Encyclopedia > Bounded

  Article Content

Bounded

The term bounded appears in different parts of mathematics where a notion of "size" can be given. The basic intuitive meaning common to all of them is that something is of finite size, and that this is the case if it is smaller than some other object that has a finite size. (Otherwise it is unbounded.) For the precise definition no precise definition of 'size' is needed.

Calculus

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.

Metric spaces

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.

Functional analysis

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.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Kings Park, New York

... and 10.6% have someone living alone who is 65 years of age or older. The average household size is 2.86 and the average family size is 3.32. In the town the ...

 
 
 
This page was created in 41.4 ms