Hermitian form can be written as sum of symmetric and alternating form

Question

Let $h:E\times E \rightarrow \mathbb{C}$ be a hermitian form.Consider the real and imaginary part of $h$ : $h(x,y)= g(x,y)+ if(x,y)$

Prove that $g$ and $f$ are bilinear,$g$ is symmetric and $f$ is alternating.

This is the problem from Serge Lang "Algebra" Chapter 10,Que 1 (a). Thanks in advance

Answer 1

Hint: Since $h$ is Hermitian, we have $$ g(y,x) + i \,f(y,x)= \overline{g(x,y) + if(x,y)} $$ where $\overline z$ denotes the complex conjugate of $z$. Note that both $f$ and $g$ are $\Bbb R$-valued.