# finite fields – Vector areas the place induced matrix norm definitions aren’t equal Answer

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## finite fields – Vector areas the place induced matrix norm definitions aren’t equal

In linear algebra, we outline an induced norm on a matrix $$A: X to X$$ by

$$||A|| = sup_{x in X setminus {0}} frac$$

and on the subsequent line we are saying it has an equal definition
$$||A|| = sup_x ||Ax||$$
as a route to embed the normalization into the clique we’re taking the supremum over.

Are there any vector areas the place the 2 definitions are not equal?

My preliminary thought was that we wanted to assemble a vector bailiwick the place there isn’t any $$x in X: ||x|| = 1$$,which eliminates any areas constructed on $$mathbb R$$ or $$mathbb C$$. A colleague advised finite fields energy proffer an instance, however that is outdoors my background.

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