Let \(V\) be an inner product space, and let \(\alpha, \beta\) be vectors in \(V\). Show that \(\alpha = \beta\) if and only if \((\alpha \mid \gamma) = (\beta \mid \gamma)\) for every \(\gamma\) in \(V\).
Solution
If \(\alpha=\beta\), then of course \[ (\alpha\mid\gamma)=(\beta\mid\gamma) \] for every \(\gamma\in V\).
Conversely, suppose \[ (\alpha\mid\gamma)=(\beta\mid\gamma) \] for every \(\gamma\). By linearity in the first argument, \[ (\alpha-\beta\mid\gamma)=0 \] for every \(\gamma\in V\). Taking \[ \gamma=\alpha-\beta \] gives \[ \|\alpha-\beta\|^2 =(\alpha-\beta\mid\alpha-\beta)=0. \] Positive definiteness therefore implies \(\alpha-\beta=0\), so \[ \boxed{\alpha=\beta.} \] This proves both directions [@HK71].