# 0708-1300/fact

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If then .

Proof

Assume that is an isomorphism. Let be the matrix of in the canonical basis and the maximum of the absolute values of the entries of . If we evaluate in all the vectors of who's entries have absolute values less than or equal to (there are of such elements) then we get elements who's entries have absolute value less than or equal to (there are of such elements in ). Since is injective we must have for every . Replacing by its inverse if necessary we can assume that but if this is the case the inequality above can not be true for arbitrarily large values of .