Verify that the standard inner product on \(F^n\) is an inner product.
Solution
Let \(F = \RR\) or \(\CC\), and define \(\inner{\alpha}{\beta} = \sum_{i=1}^n \alpha_i \overline{\beta_i}\) for \(\alpha = (\alpha_1, \ldots, \alpha_n)\), \(\beta = (\beta_1, \ldots, \beta_n) \in F^n\).
1. Conjugate symmetry: \(\inner{\alpha}{\beta} = \overline{\inner{\beta}{\alpha}}\).
\(\inner{\alpha}{\beta} = \sum \alpha_i \overline{\beta_i} = \overline{\sum \beta_i \overline{\alpha_i}} = \overline{\inner{\beta}{\alpha}}\).
2. Linearity in the first argument: \(\inner{c\alpha + \beta}{\gamma} = c\inner{\alpha}{\gamma} + \inner{\beta}{\gamma}\).
\(\sum (c\alpha_i + \beta_i)\overline{\gamma_i} = c\sum \alpha_i \overline{\gamma_i} + \sum \beta_i \overline{\gamma_i}\).
3. Positive definiteness: \(\inner{\alpha}{\alpha} \ge 0\), with equality iff \(\alpha = 0\).
\(\inner{\alpha}{\alpha} = \sum |\alpha_i|^2 \ge 0\), and this is \(0\) iff every \(\alpha_i = 0\).
4. Hence the standard inner product satisfies the three axioms of an inner product.
<1>1–<1>3.
5. Q.E.D.
4.