About a property of inner product space

Question

I don't understand a property of inner product space: $$\langle a,b\rangle=\overline{\langle b,a \rangle}$$ Let $a=1+i, b=2+2i$, then $\langle a,b\rangle=(1+i)(2+2i)=2+4i-2$ and $\overline{\langle b,a\rangle }= (2-2i)(1-i)=2-4i-2$.

How are they equal?

Answer 1

The inner product on complex numbers is $\overline{a}b$ (or $a\overline{b}$; you'll have to forgive me, I'm a physicist) rather than $ab$, but in any case you've computed a product of conjugates, not the conjugate of a product. Note $\overline{a}b=\overline{\overline{b}a}$.

Answer 2

There is some confusion here. The inner product in $\mathbb C$ is $\langle z,w\rangle=z\overline w$. And, with this inner product, you can check that indedd$$\langle 1+i,2+2i\rangle=\overline{\langle2+2i,1+i\rangle}.$$