∣α+β∣2+∣α−β∣2=2(∣α∣2+∣β∣2)|\alpha + \beta|^2 + |\alpha - \beta|^2 = 2(|\alpha|^2 + |\beta|^2)∣α+β∣2+∣α−β∣2=2(∣α∣2+∣β∣2) 左辺 = (α+β)(α+β‾)+(α−β)(α−β‾)(\alpha + \beta)(\overline{\alpha + \beta}) + (\alpha - \beta)(\overline{\alpha - \beta})(α+β)(α+β)+(α−β)(α−β) =αα‾+αβ‾+α‾β+ββ‾+αα‾−αβ‾−α‾β+ββ‾= \alpha\overline{\alpha} + \alpha\overline{\beta} + \overline{\alpha}\beta + \beta\overline{\beta} + \alpha\overline{\alpha} - \alpha\overline{\beta} - \overline{\alpha}\beta + \beta\overline{\beta}=αα+αβ+αβ+ββ+αα−αβ−αβ+ββ =∣α∣2+∣β∣2+∣α∣2+∣β∣2= |\alpha|^2 + |\beta|^2 + |\alpha|^2 + |\beta|^2=∣α∣2+∣β∣2+∣α∣2+∣β∣2 =2(∣α∣2+∣β∣2)== 2(|\alpha|^2 + |\beta|^2) = =2(∣α∣2+∣β∣2)= 右辺 よって与式は成り立つ