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I've been trying to prove that the following is a norm, but wasn't successful. I also cannot find a counterexample. So help is greatly appreciated. Let $x \in \mathbb{R}^N, \ w_i \in \mathbb{R}_+,\ i=1,\ldots,N$. $$\lVert x \rVert_w := \max \lvert w_i x_i\rvert$$

Basically, this is the maximum norm with positive weights assigned to each dimension.

It must be shown that: $$\max \lvert w_i (x_i+y_i) \rvert \leq \max \lvert w_j x_j \rvert + \max \lvert w_k y_k \rvert$$

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If you assume that $\mathbb{R}_+=[0,+\infty)$ then $\Vert x\Vert_w$ is not a norm, just a seminorm. –  Norbert Aug 13 '12 at 13:16
    
@Norbert: (quote) with positive weights assigned to each dimension. –  Did Aug 13 '12 at 13:25

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up vote 3 down vote accepted

Since $|w_ix_i|\leqslant \lVert x \rVert_w$ and $|w_iy_i|\leqslant \lVert y \rVert_w$ for every $i$, $$ |w_i(x_i+y_i)|\leqslant|w_ix_i|+|w_iy_i|\leqslant\lVert x\rVert_w+ \lVert y \rVert_w, $$ for every $i$. This proves that $\max\limits_i|w_i(x_i+y_i)|\leqslant\lVert x\rVert_w+ \lVert y \rVert_w$, QED.

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