Alternative Definitions
If the vector norm is an inner product norm, as in a Hilbert space, then the logarithmic norm is the smallest number such that for all
Unlike the original definition, the latter expression also allows to be unbounded. Thus differential operators too can have logarithmic norms, allowing the use of the logarithmic norm both in algebra and in analysis. The modern, extended theory therefore prefers a definition based on inner products or duality. Both the operator norm and the logarithmic norm are then associated with extremal values of quadratic forms as follows:
Read more about this topic: Logarithmic Norm
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