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.
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