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If <math>B</math> is a bounded neighborhood of the origin in a (locally bounded) TVS then its image under any continuous linear map will be a bounded set (so this map is thus bounded on this neighborhood <math>B</math>).
Consequently, a linear map from a locally bounded TVS into any other TVS is continuous if and only if it is [[#bounded on a neighborhood|bounded on a neighborhood]].
Moreover, any TVS with this property must be a locally bounded TVS. Explicitly, if <math>X</math> is a TVS such that every continuous linear map (into any TVS) whose ___domain is <math>X</math> is necessarily bounded on a neighborhood, then <math>X</math> must be a locally bounded TVS (because the [[identity function]] <math>X \to X</math> is
Any linear map from a TVS into a locally bounded TVS (such as any linear functional) is continuous if and only if it is bounded on a neighborhood.{{sfn|Wilansky|2013|pp=53-55}}
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