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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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$\|x\|_p=\left(\sum_{i=1}^n|x_i|^p\right)^{1/p}$. –  GWu Apr 12 '11 at 17:58
    
Checking that this is a norm is then quite straightforward. What about the next step? –  Libertron Apr 12 '11 at 18:02
    
Holder and Minkovski? This is the simple case that should be proved before the corresponding versions about functions. –  GWu Apr 12 '11 at 18:08
    
I guess you can find proof online. For instance, math.ksu.edu/~nagy/real-an/ap-d-holder.pdf –  GWu Apr 12 '11 at 18:09
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One way to do homework is ``google" :) –  GWu Apr 12 '11 at 18:12
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