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Let $J=[a,b] \subset \mathbb{R}$, the slave (me) is asked to show that :

$$u\in C(J,\mathbb{R}^n) \mapsto ||u||_{1} = \sum_{i=1}^{n} ||u_i||_{\infty} = \sum_{i=1}^{n}max_{x\in J}|u_i(x)|$$

and $$u\in C(J,\mathbb{R}^n) \mapsto ||u||_{\infty} = max_{x\in J} || u(x)||_{1} = max_{x\in J } \sum_{i=1}^{n} |u_i (x)|$$

are equivalent ($u_i \in C(J,\mathbb{R})$ are components of u)

I must find $c_1, c_2 $ which will satisfy: $$ c_{1} ||u||_\infty \le ||u||_1 \le c_2 ||u||_\infty$$

Maybe : $u_i(x) = u_1(x),......,u_n(x)$ and so : $|u_i (x)| = |u_1 (x)|,...,|u_n (x)|$

and then: $max (|u_i (x)| )= u_{m}(x_{max}) = m $ so : $||u||_{1} = \sum m = m $

but, for $||u||_{\infty}$ we get: $||u||_\infty = max(|u_1(x)|+|u_2(x)| +... + |u_n(x)| ) $

so : $ ||u||_1 \le ||u||_\infty$

so they must be equivalent.

I have a feeling this is wrong, forgive me and point me to the right way please...

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1. You have assumed one form of $u(x)$ where all the components $u_i(x)$ are equal, but the question is to prove it for all $u(x)$. 2. You have argued that $\lVert u\rVert_1\le\lVert u\rVert_\infty$, but you also have to show that $\lVert u\rVert_1\ge c_1\lVert u\rVert_\infty$ for some fixed $c_1$. –  Rahul Feb 3 '13 at 20:30
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