Definition10.1Norm on \(\Cm\)
A function \(\| \cdot \|\) is called a norm on \(\Cm\) if, for all vectors \(x\) and \(y\) in \(\Cm\)
- \(\|x\| \geq 0\text{,}\) and \(\|x\|=0\) if and only if \(x=0\text{.}\)
- \(\|\lambda x\| = |\lambda| \| x\|\) for any scalar \(\lambda \in \Cs\text{.}\)
- \(\|x + y\| \leq \| x\| + \| y\|\) (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
- The 1-norm (or “taxi-cab norm”): \(\|\vec x\|_1 = \sum_{i=1}^n |x_i|\text{.}\)
- 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 \(x^\star x = \|x\|_2^2
\text{.}\)
- 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.
Lemma10.3Cauchy-Schwarz inequality
For all \(x, y \in \Cm\text{,}\)
\begin{equation*}
| x^Ty | \leq \| x\|_2 \| y \|_2.
\end{equation*}
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.