# Section10Vector Norms (NM, 25 Sep)¶ permalink

##### Definition10.1Norm on $\Cm$

A function $\| \cdot \|$ is called a norm on $\Cm$ if, for all vectors $x$ and $y$ in $\Cm$

1. $\|x\| \geq 0\text{,}$ and $\|x\|=0$ if and only if $x=0\text{.}$
2. $\|\lambda x\| = |\lambda| \| x\|$ for any scalar $\lambda \in \Cs\text{.}$
3. $\|x + y\| \leq \| x\| + \| y\|$ (triangle inequality).

The norm of a vector gives us some information about the “size” of the vector. But there are different ways of measuring the size: you could take the absolute value of the largest entry, you cold look at the “distance” for the origin, etc...

##### Definition10.2$p$-norms.

For any $p \lt 1\text{,}$ define

\begin{equation*} \|x\|_p = \big( \sum_{i=1}^m|x_i|^p\big)^{1/p}. \end{equation*}

Of particular importance are the norms

1. The 1-norm (or “taxi-cab norm”): $\|\vec x\|_1 = \sum_{i=1}^n |x_i|\text{.}$
2. The $2$-norm (a.k.a. the Euclidean norm): $\|x\|_2 = \bigg(\sum_{i=1}^n |x_i|^2\bigg)^{1/2}.$ Note, if $x \in \Cm \text{,}$ then $x^\star x = \|x\|_2^2 \text{.}$
3. The $\infty$-norm (also known as the max-norm): $\|\vec x\|_\infty = \max_{i=1}^n |x_i|\text{.}$

It takes a little bit of effort to show that $\| \cdot \|_2$ satisfies the triangle inequality. First we need the Cauchy-Schwarz inequality.

This can then be applied to show that

\begin{equation*} \| x + y \|_2 \leq \| x\|_2 + \|y\|_2. \end{equation*}

It follows directly that $\| \cdot\|_2$ is a norm.

# Subsection10.1Exercises

##### Exercise10.4

Define the function $\| \cdot \|_{A,1} : \Cm \to \R$ as $\|x\|_{A,1} = \|Ax\|_1$ where $\|\cdot \|_1$ is the usual 1-norm (a.k.a. “the taxicab norm”) on $\Cm\text{,}$ and $A \in \Cmm$ is non-singular. Is it true that $\|Ax\|_1$ is a norm on $\Cm\text{?}$

##### Exercise10.5

Show that the following inequalities hold for all vectors $x \in \Cm\text{.}$ If possible, give a nontrivial example for which equality holds.

1. $\|x\|_\infty \leq \| x\|_2$
2. $\|x\|_2 \leq \sqrt{m}\|x\|_\infty$
3. $\|x\|_2 \leq \sqrt{\|x\|_1 \|x\|_\infty}$
4. $\|x\|_\infty \leq \|x\|_2 \leq \|x\|_1$

Which, if any, of these inequalities extend to the matrix norms induced by these vector norms?