We introduce and study an asymptotic norm of an operator as the infimum of the norms of its restrictions to all finite codimensional subspaces. The main result asserts that the asymptotic norm of an operator equals to the distance of the operator from the set of all compact (equivalently, finitedimensional) operators with wider range spaces. On the other hand, we prove that the asymptotic norm is "close" to the measure of non-compactness.

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