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

Jaylyn Gibson

Answered question

2022-08-07

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?

Answer & Explanation

a2t2esdg

a2t2esdg

Beginner2022-08-08Added 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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