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If a TVS happens to also be a normed (or seminormed) space then a subset <math>S</math> is von Neumann bounded if and only if it is norm bounded; that is, if and only if <math>\sup_{s \in S} \|s\| < \infty.</math>
If <math>S \subseteq X</math> is a set then <math>F : X \to Y</math> is said to be {{em|bounded on <math>S</math>}} if <math>F(S)</math> is a [[Bounded set (topological vector space)|bounded subset]] of <math>Y,</math> and it is said to be {{em|unbounded on <math>S</math>}} otherwise.
If <math>F</math> is a valued in a normed (or seminormed) space <math>(Y, \|\cdot\|),</math> such as the scalar field (<math>\R</math> or <math>\Complex</math>) for instance, then it is bounded on a subset <math>S</math> if and only if <math>\sup_{s \in S} \|F(s)\| < \infty.</math>
A linear map <math>F</math> is bounded on a set <math>S</math> if and only if it is bounded on <math>x + S</math> for every <math>x \in X</math> (because <math>F(x + S) = F(x) + F(S)</math>).
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