How do I find a constant B>0, so that norm(x) <= B * norm(x)_oo works for all x in RR^n?

Jaylyn Gibson 2022-08-07 Answered
How do I find a constant B>0, so that x B x works for all x R n ?
Not sure but I think I have to look at x = i = 1 n x i e i with the development of the x to the canonical unit vectors e i . But how do I do that?
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Answers (1)

a2t2esdg
Answered 2022-08-08 Author has 13 answers
Write x = x 1 e 1 + + x n e n . If is an arbitrary norm on R n you have
x = x 1 e 1 + x n e n i = 1 n x i e i = i = 1 n | x i | e i .
Each x i satisfies | x i | x so that
x = x 1 e 1 + x n e n i = 1 n x e i = ( i = 1 n e i ) x .
Take
B = i = 1 n e i .
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