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With the following norm inequality:

||Av|| ≤ ||A||||v|| implies ||A|| = sup

_{v}[ ||Av||/||v|| ]

I understand that sup is the upper bound of a set B, or least upper bound if B is a subset of A, where the upper bounds are elements of both B and A.

Is this saying that the norm of A is the maximum of the set ||Av||/||v||, where there are multiple vectors v being considered?