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Mathematics LibreTexts

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
    \norm{\cdot} : V &\to \mathbb{R}\\
    v &\mapsto \norm{v}
    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
    \begin{equation} \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}).