Hermitian Spaces

Abstract

Recall that a vector space W over the field $\mathbb{C}$ is called Hermitian if for any two vectors u, w ∈ W, the $\mathbb{R}$-bilinear Hermitian inner product $(u,w) \in \mathbb{C}$ is defined such that

$$\displaystyle{(u,w) = \overline{(w,u)}\, ,\quad (zu,w) = z\,(u,w) = (u,\overline{z}w)\, ,\quad \text{and}\quad (w,w) > 0\text{ for all }w\neq 0\,.}$$

It provides every vector w ∈ W with a Hermitian norm $\Vert w\Vert \stackrel{\text{def}}{=}\sqrt{(w, w)} \in \mathbb{R}_{\geqslant 0}$. Since

$$\displaystyle\begin{array}{rcl} (u + w,u + w)& =& \Vert u\Vert ^{2} +\Vert w\Vert ^{2} + 2\,\mathop{\mathrm{Re}}\nolimits (u,w), {}\\ (u + iw,u + iw)& =& \Vert u\Vert ^{2} +\Vert w\Vert ^{2} - 2i\,\mathop{\mathrm{Im}}\nolimits (u,w)\, , {}\\ \end{array}$$

the Hermitian inner product is uniquely recovered from the norm function and the multiplication-by-i operator as

$$\displaystyle{ 2(w_{1},w_{2}) =\Vert w_{1} + w_{2}\Vert ^{2} -\Vert w_{ 1} + iw_{2}\Vert ^{2}\,. }$$

Note that this agrees with the general ideology of Kähler triples from Sect. 18.5.2 on p. 471.