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.

Positive definiteness: $\norm{v}=0 $ if and only if $v=0$;
Positive Homogeneity: $\norm{av}=|a|\,\norm{v} $ for all $a\in \mathbb{F} $ and $v\in V$;
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
\begin{equation} \label{eqn:NormInRn}
\norm{v} = \sqrt{x_1^2+\cdots + x_n^2}. \tag{9.2.2}
\end{equation}

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

$A vector in R3 final.jpg$

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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