How do you prove the $p$-norm is not a norm in $\mathbb R^n$ when $0<p<1$?

real-analysis examples-counterexamples normed-spaces lp-spaces

Consider $(1,0,0,\ldots, 0)+(0,1,0,\ldots,0) = (1,1,0,\ldots,0)$.


Yes, a short counter-example.

assume that $n=2$,take vectors $(1,0)$ and $(0,1)$.

you'll find it doesn't satisfy triangle inequality.

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