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In other words, the commutators must lie in the linear span of the at every point.
en.wikipedia.org
Let and denote the complex linear span of and respectively.
en.wikipedia.org
It then immediately follows that the range of is the linear span of these product states.
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Perhaps the best known example for this is the function that associates to every subset of a given vector space its linear span.
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Every vector in the new set is orthogonal to every other vector in the new set; and the new set and the old set have the same linear span.
en.wikipedia.org

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