Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
$\|x\|_p=\left(\sum_{i=1}^n|x_i|^p\right)^{1/p}$. – 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, – GWu Apr 12 '11 at 18:09
One way to do homework is ``google" :) – GWu Apr 12 '11 at 18:12

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.