5 Normed Vector Spaces
5.1 Definition of a Normed Vector Space
A Normed Vector Space \(\mathcal{N}\) is a vector space \(\mathcal{V}\) over a field \(\mathcal{F}\) equipped with a Norm \(\left\Vert \cdot\right\Vert :\mathcal{V}\rightarrow\mathbb{R}\) obeying the following axioms.
\(\forall|a\rangle,|b\rangle\in V\); \(\alpha\in F:\)
[N1]: \(\left\Vert |a\rangle\right\Vert \ge0\) (non-negativity)
[N2]: \(\left\Vert |a\rangle\right\Vert =0\implies|a\rangle=|0\rangle\) (positive definiteness)
[N3]: \(\left\Vert \alpha|a\rangle\right\Vert =\left|\alpha\right|\left\Vert |a\rangle\right\Vert\) (absolute homogeneity)
[N4]: \(\left\Vert |a\rangle+|b\rangle\right\Vert \le\left\Vert |a\rangle\right\Vert +\left\Vert |b\rangle\right\Vert\) (subadditivity, aka the triangle inequality).
Equivalently, a normed vector space is an ordered pair \(\mathcal{N}=\left(\mathcal{V},\left\Vert \cdot\right\Vert \right)\) obeying these axioms.
5.2 Examples of Normed Vector Spaces
Lemma [\(l\)I2] implies that we can always define the norm in any inner product space:
For a vector \(|a\rangle\) in an inner product space \(\mathcal{I}\), the Norm induced by the inner product is defined to be
\[\left\Vert |a\rangle\right\Vert \triangleq\sqrt{\langle a|a\rangle}\].
Hence, every inner product space is also a normed vector space: \(\mathcal{I}\subseteq\mathcal{N}\). As a result, all the examples of inner product spaces serve as examples of normed vector spaces.