I previously claimed that the norm of a vector in a space with a defined inner product is \begin{equation} \Vert \vb{a} \Vert = \sqrt{\ev{\vb{a},\vb{a}}} \label{eq:spnorm} \end{equation} and this is indeed a satisfactory form. However, other definitions are possible, consistent with the following properties:
\(||\vb{v}|| \ge 0\) ; \(||\vb{v}|| = 0\) if and only if \(\vb{v} = \vb{0}\).
\(\Vert\alpha \mathbf{v}\Vert =\) \(|\alpha| \Vert\vb{v}\Vert\).
\(\Vert\vb{v}_1 + \vb{v}_2\Vert \le \Vert\vb{v}_1\Vert + \Vert\vb{v}_2\Vert\) (the triangle inequality).
\begin{align} \Vert (a_1, \ldots, a_n) \Vert &= \sqrt{ \sum_{k=1}^n |a_k|^2} \\ \Vert (a_1, \ldots, a_n) \Vert &= \sum_{k=1}^n |a_k| \label{eq:USPS} \\ \Vert (a_1, \ldots, a_n) \Vert &= \text{max}_{k=1}^n |a_k| \\ \Vert (a_1, \ldots, a_n) \Vert &= \text{max}_{k=1}^n k|a_k| \end{align}
You may confirm that each of these definitions satisfies all three required properties to be a norm. As Nearing points out, “the United States Postal Service prefers a variation on Eq. (\ref{eq:USPS}), which makes good sense in a place like Manhattan, though not as much in Boston.
A scalar product is a scalar function of two vectors with the following properties:
\(\ev{\vb{w},(\vb{u} + \vb{v})} = \ev{\vb{w}, \vb{u}} + \ev{\vb{w}, \vb{v}}\).
\(\ev{\vb{w}, \alpha\vb{v}} = \alpha \ev{\vb{w}, \vb{v}}\).
\(\ev{\vb{u}, \vb{v}}^{*} = \ev{\vb{v}, \vb{u}}\).
\(\ev{\vb{v}, \vb{v}} \ge 0;\quad\text{and}\quad \ev{\vb{v}, \vb{v}} = 0\) if and only if \(\vb{v} = \vb{0}\).
If a scalar product is defined on a particular vector space, then a natural definition for norm is the one we started with in Eq. (\ref{eq:spnorm}). However, consistency requires us to show that this norm satisfies the triangle inequality.
We argued that cubic polynomials on \(0 \le x \le 1\) form a vector space, in which the basis vectors could be taken to be \((1, x, x^2, x^3)\), the addition of vectors and scalar multiplication are as expected: \begin{align} (a_0 + a_1 x + a_2 x^2 + a_3 x^3) + (b_0 + b_1 x + b_2 x^2 + b_3 x^3) &= (a_0 + b_0) + (a_1 + b_1) x + (a_2 + b_2) x^2 + (a_3 + b_3) x^3 \\ \alpha (a_0 + a_1 x + a_2 x^2 + a_3 x^3) &= (\alpha a_0) + (\alpha a_1) x + (\alpha a_2) x^2 + (\alpha a_3) x^3 \end{align}
What could be a scalar product for this vector space? Consider \begin{equation} \ev{f, g} = \int_0^1 f(x)^* g(x) \dd{x} \end{equation} Does it satisfy all four required properties for a scalar product? Can you define a norm using this definition?
The Cauchy-Schwartz inequality is \begin{equation} \label{eq:cauchy-schwartz} |\ev{\vb{u}, \vb{v}}| \le \Vert\vb{u}\Vert \Vert\vb{v}\Vert \end{equation} You can prove it by considering \(\ev{\vb{u} - \lambda \vb{v}, \vb{u} - \lambda \vb{v}} \ge 0\), where \(\lambda\) is an arbitrary scalar.