A calculus of the absurd

20.8.3 Induced norms

Every norm “induces” an inner product, which is to say that if we have an inner product, we can define the norm

\begin{equation} \norm {\mathbf {x}} = \sqrt {\langle \mathbf {x}, \mathbf {x} \rangle } \end{equation}

To show that this actually is a norm it is necessary and sufficient to show that it satisfies the four norm axioms.

  • Homogeneity. Consider \(\norm {\alpha \mathbf {x}}\), for which

    \begin{align} \norm {\alpha \mathbf {x}} &= \sqrt {\langle \alpha \mathbf {x}, \alpha \mathbf {x} \rangle } \\ &= \sqrt {\alpha \langle \mathbf {x}, \alpha \mathbf {x} \rangle } \\ &= \sqrt {\alpha \overline {\langle \alpha \mathbf {x}, \mathbf {x} \rangle }} \\ &= \sqrt {\alpha \overline {\alpha } \overline {\langle \mathbf {x}, \mathbf {x} \rangle }} \\ &= \sqrt {\abs {\alpha } \langle \mathbf {x}, \mathbf {x} \rangle } \end{align}

  • TODO: other proofs