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A subset of a normed (or seminormed) space is called {{em|bounded}} if it is norm-bounded (or equivalently, von Neumann bounded).
For example, the scalar field (<math>\Reals</math> or <math>\Complex</math>) with the [[absolute value]] <math>|\cdot|</math> is a normed space, so a subset <math>S</math> is bounded if and only if <math>\sup_{s \in S} |s|</math> is finite, which happens if and only if <math>S</math> is contained in some open (or closed) ball centered at the origin (zero).
Any translation and scalar multiple of a bounded set is again bounded.
'''Function bounded on a set'''
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