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In [[mathematics]], [[linear map]]s form an important class of "simple" [[function (mathematics)|functions]] which preserve the algebraic structure of [[linear space]]s and are often used as approximations to more general functions (see [[linear approximation]]). If the spaces involved are also [[topological space]]s (that is, [[topological vector space]]s), then it makes sense to ask whether all linear maps are [[continuous map|continuous]]. It turns out that for maps defined on infinite-[[dimension (linear algebra)|dimensional]] topological vector spaces (e.g., infinite-dimensional [[normed space]]s), the answer is generally no: there exist '''discontinuous linear maps'''. If the ___domain of definition is [[complete space|complete]], it is trickier; such maps can be proven to exist, but the proof relies on the [[axiom of choice]] and does not provide an explicit example.
== A linear map from a finite-dimensional space is always continuous ==
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