Let $X=K^3$. For $x=(x(1),x(2),x(3))\in X$, let $||x||=[(|x(1)|^2+|x(2)|^2)^\frac{3}{2}+|x(3)|^3]^\frac{1}{3}$. Then $||.||$ is a norm on $K^3$
How can we show that the following is a norm
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functional-analysis