I am working with complex numbers ($\mathbb {C} $).
I'm trying to figure this out:
$ \forall x,y\in V,\ \langle x,y\rangle =\overline {\langle y,x\rangle }$
When we talk about inner product space do I need to multiply between the vectors?
like normal multiplication between vectors? (only with complex numbers)
If anyone can give me an example for ($ \forall x,y\in V,\ \langle x,y\rangle =\overline {\langle y,x\rangle }$) (but with numbers to understand) it can greatly help.
Thank you.
Complex numbers form a $1$-dimensional vector space over themselves with inner product $\langle x,\,y\rangle:=x\overline{y}$. We can also regard this as a $2$-dimensional space over the reals. Writing $x=a+bi, \,y=c+di$ gives $x\overline{y}=ac+bd+(bc-ad)i$. Either choice of scalars lets you verify $\langle x,\,y\rangle=\overline{\langle y,\,x\rangle}$.