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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 <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>
By definition, a linear map <math>F : X \to Y</math> between [[Topological vector space|TVS]]s is said to be {{em|[[Bounded linear operator|bounded]]}} and is called a {{em|[[bounded linear operator]]}} if for every [[Bounded set (topological vector space)|(von Neumann) bounded subset]] <math>B \subseteq X</math> of its ___domain, <math>F(B)</math> is a bounded subset of it codomain; or said more briefly, if it is bounded on every bounded subset of its ___domain.
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* By definition of the set <math>U^{\circ},</math> which is called the [[Polar set|(absolute) polar]] of <math>U,</math> the inequality <math>\sup_{u \in U} |f(u)| \leq 1</math> holds if and only if <math>f \in U^{\circ}.</math> Polar sets, and thus also this particular inequality, play important roles in [[duality theory]].
</li>
<li><math>f</math> is a [[#locally bounded|locally bounded at every point]] of its ___domain.</li>
▲* Moreover, <math>f</math> is bounded on a set <math>U</math> if and only if <math>f</math> is bounded on <math>x + U</math> for every <math>x \in X</math> (because <math>f(x + U) = f(x) + f(U)</math>).</li>
<li>The kernel of <math>f</math> is closed in <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=156-175}}</li>
<li>Either <math>f = 0</math> or else the kernel of <math>f</math> is {{em|not}} dense in <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=156-175}}</li>
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