For a point $x = (x_1, x_2,\ldots, x_n)$ in $\mathbb R^n$, define $T_x$ to be the step function on the interval $\left[1, n+1\right)$ that takes the value $x_k$ on the interval $\left[k, k+1\right)$ for $1\leq k$ $\leq$ $n$. For $p\geq1$, define $\lVert x\rVert_p = \lVert T_x\rVert_p$, the norm of the function $T_x$ in $L^p(\left[1, n+1\right))$. How do we show that this defines a norm on $\mathbb R^n$? How would we prove the Hölder and Minkowski Inequalities for this norm?
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