Prove some properties of a new definition of the norm

by RoRi

April 28, 2016

Consider an alternative definition for the norm of a vector $A = (a_1, \ldots, a_n)$ given by

$\lVert A \rVert = \max_{1 \leq k \leq n} | a_k |.$

Which properties of Theorems 12.4 and 12.5 (pages 453-454 of Apostol) are still valid with this new definition?
Draw a figure which shows the set of points $(x,y) \in \mathbb{R}^2$ of norm 1 with this new definition.

Claim. All of the properties of Theorems 12.4 and 12.5 hold with this new definition.
Proof. For Theorem 12.4(a), if $A \neq O$ then there is some $a_i \neq 0$ so $\max_{1 \leq k \leq n} |a_k| > 0$ . Therefore $\lVert A \rVert > 0$ .

For Theorem 12.4(b), if $A = O$ then $a_i = 0$ for all $1 \leq i \leq n$ . Therefore $\max_{1 \leq k \leq n} |a_k| = 0$ and so $\lVert A \rVert = 0$ .

For Theorem 12.4(c), we compute

$\begin{align*} \lVert cA \rVert &= \max_{1 \leq k \leq n} |ca_k| \\ &= \max_{1 \leq k \leq n} |c| |a_k| \\ &= |c| \max_{1 \leq k \leq n} |a_k| \\ &= |c| \lVert A \rVert. \end{align*}$

For Theorem 12.5 we have

$\begin{align*} \lVert A + B \rVert &= \max_{1 \leq k \leq n} | a_k + b_k | \\ &\leq \max_{1 \leq k \leq n} \left( |a_k| + |b_k| \right) \\ &\leq \max_{1 \leq k \leq n} |a_k| + \max_{1 \leq k \leq n} |b_k| \\ &= \lVert A \rVert + \lVert B \rVert. \qquad \blacksquare \end{align*}$
We have the following figure,

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