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Indeed, this example shows that every [[Locally convex topological vector space|locally convex space]] that is not seminormable has a linear TVS-[[automorphism]] that is not bounded on any neighborhood of any point.
Thus although every linear map that is bounded on a neighborhood is necessarily continuous, the converse is not guaranteed in general.
'''Guaranteeing "bounded" implies "bounded on a neighborhood"'''▼
If <math>F : X \to Y</math> is a bounded linear operator from a [[normed space]] <math>X</math> into some TVS then <math>F : X \to Y</math> is necessarily continuous; this is because any open ball <math>B</math> centered at the origin in <math>X</math> is both a bounded subset (which implies that <math>F(B)</math> is bounded since <math>F</math> is a bounded linear map) and a neighborhood of the origin in <math>X,</math> so that <math>F</math> is thus bounded on this neighborhood <math>B</math> of the origin, which (as mentioned above) guarantees continuity.▼
'''Guaranteeing "bounded" implies "continuous"'''
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A linear map from a TVS into a [[Normed space|normed]] or [[Seminormed space|seminormed]] space (such as a linear functional for example) is continuous if and only if it is bounded on some neighborhood. In particular, a linear functional on a arbitrary TVS is continuous if and only if it is bounded on some neighborhood.
In addition, a linear map from a [[Normed space|normed]] or [[Seminormed space|seminormed]] space into a TVS is continuous if and only if it is bounded on a neighborhood. Thus when the ___domain or codomain of a linear map is normable or seminormable, then continuity is equivalent to being bounded on a neighborhood.
▲'''Guaranteeing "bounded" implies "bounded on a neighborhood"'''
▲If <math>F : X \to Y</math> is a bounded linear operator from a [[normed space]] <math>X</math> into some TVS then <math>F : X \to Y</math> is necessarily continuous; this is because any open ball <math>B</math> centered at the origin in <math>X</math> is both a bounded subset (which implies that <math>F(B)</math> is bounded since <math>F</math> is a bounded linear map) and a neighborhood of the origin in <math>X,</math> so that <math>F</math> is thus bounded on this neighborhood <math>B</math> of the origin, which (as mentioned above) guarantees continuity.
=== Properties of continuous linear operators ===
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