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== A concrete example ==
Examples of discontinuous linear maps are easy to construct in spaces that are not complete; on any Cauchy sequence of independent vectors which does not have a limit, a linear operator may grow without bound.{{what|reason=This is not a proof nor even clear statement of anything, yet later in the article it is treated as an established principle.}} In a sense, the linear operators are not continuous because the space has "holes".
For example, consider the space ''X'' of real-valued [[smooth function]]s on the interval [0, 1] with the [[uniform norm]], that is,
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