
# 9.2 Norms

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The norm of a vector in an arbitrary inner product space is the analog of the length or magnitude of a vector in $$\mathbb{R}^{n}$$. We formally define this concept as follows.

Definition 9.2.1.  Let $$V$$ be a vector space over $$\mathbb{F}$$. A map
\begin{equation*}
\begin{split}
\norm{\cdot} : V &\to \mathbb{R}\\
v &\mapsto \norm{v}
\end{split}
\end{equation*}
is a norm on $$V$$ if the following three conditions are satisfied.

1. Positive definiteness: $$\norm{v}=0$$ if and only if $$v=0$$;
2. Positive Homogeneity: $$\norm{av}=|a|\,\norm{v}$$ for all $$a\in \mathbb{F}$$ and $$v\in V$$;
3. Triangle inequality: $$\norm{v+w}\le \norm{v}+\norm{w}$$ for all $$v,w\in V$$.

Remark 9.2.2.  Note that, in fact, $$\norm{v}\ge 0$$ for each $$v\in V$$ since

$0 = \norm{v-v} \le \norm{v} + \norm{-v} = 2\norm{v}.$

Next we want to show that a norm can always be defined from an inner product $$\inner{\cdot}{\cdot}$$ via the formula

$\norm{v} = \sqrt{\inner{v}{v}} ~\text{for all} ~ v \in V . \tag{9.2.1}$

Properties 1 and 2 follow easily from Conditions~1 and 3 of Definition~9.1.1. The triangle inequality requires more careful proof, though, which we give in Theorem~9.3.4\ref{thm:triangle} in the next chapter.

If we take $$V=\mathbb{R}^n$$, then the norm defined by the usual dot product is related to the usual notion of length of a vector. Namely, for $$v=(x_1,\ldots,x_n)\in \mathbb{R}^n$$, we have
\label{eqn:NormInRn}
\norm{v} = \sqrt{x_1^2+\cdots + x_n^2}.      \tag{9.2.2}

We illustrate this for the case of $$\mathbb{R^3}$$ in Figure 9.2.1.

Figure 9.2.1: The length of a vector in $$\mathbb{R^3}$$ via equation 9.2.1.

While it is always possible to start with an inner product and use it to define a norm, the converse requires more care.  In particular, one can prove that a norm can be used to define an inner product via Equation 9.2.1 if and only if the norm satisfies the Parallelogram Law (Theorem 9.3.6~\ref{thm:ParallelogramLaw}).

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