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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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