I'm thoroughly confused by this question:
Let $\langle\,,\rangle$ be a positive definite Hermitian form on a complex vector space $V$, and let $\left\{ \ , \ \right\}$ and $[\ ,\ ]$ be its real and imaginary parts: $\langle v,w \rangle = \left\{ v , w \right\} + i[ v,w ]$. Prove that when $V$ is viewed as an $\mathbb{R}$ vector space $\left\{ \ ,\ \right\}$ is a positive definite symmetric form, and $[ \ , \ ]$ is a skew-symmetric form.
So I need to show $\left\{v, w \right\} = {\left\{w,v \right\}}$ and $[v,w] = -{[w,v]}$.
I really have no idea how to go about showing this though, any help greatly appreciated.
Best Answer
Since $\langle\cdot,\cdot\rangle$ is Hermitian, we know that $\langle v,w\rangle=\overline{\langle w,v\rangle}$. Therefore $$ \langle v,w\rangle=\{v,w\}+i[v,w]=\overline{\{w,v\}+i[w,v]}=\{w,v\}-i[w,v]. $$