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Minkowski's inequaity states that $$\displaystyle{(|x_1+y_1|^p + |x_2+y_2|^p +\dots +|x_n+y_n|^p)^{\frac{1}{p}}\leq (|x_1|^p + |x_2|^p +\dots +|x_n|^p)^{\frac{1}{p}}+(|y_1|^p + |y_2|^p +\dots +|y_n|^p)^{\frac{1}{p}}}$$ for all $p\in\Bbb{N}$.

On reading the proof, I feel this can easily be generalised to $$\displaystyle{(|a_1+b_1+c_1+\dots r_1|^p + |a_2+b_2+\dots r_2|^p +\dots +|a_n+b_n+c_n+\dots r_n|^p)^{\frac{1}{p}}\leq (|a_1|^p + |a_2|^p +\dots +|a_n|^p)^{\frac{1}{p}}+(|b_1|^p + |b_2|^p +\dots +|b_n|^p)^{\frac{1}{p}}+(|c_1|^p + |c_2|^p +\dots +|c_n|^p)^{\frac{1}{p}}+\dots +(|r_1|^p + |r_2|^p +\dots +|r_n|^p)^{\frac{1}{p}}}$$ for all $p\in\Bbb{N}$.

  1. Is this generalisation correct?
  2. Is it trivial?

Thanks in advance!

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Yes, and yes. It's the subadditivity of the $\|\cdot \|_p$ norm. Once you have $\|a + b\| \le \|a\| + \|b\|$ for all $a, b$, then $$\|a + b + c\| = \|(a + b) + c\| \le \|a + b\| + \|c\| \le \|a\| + \|b\| + \|c\|$$ etc.

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As noted, the generalization is correct. At a more basic level, it follows easily by induction on the number of terms in the absolute value signs.

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