How is the basis $\{\frac{1}{\sqrt{2}}(1, i),\frac{1}{\sqrt{2}}(1, -i)\} $ orthonormal

Question

How are the bases $\frac{1}{\sqrt{2}}(1, i)$ and $\frac{1}{\sqrt{2}}(1, -i)$ orthonormal? I know for a basis to be orthonormal it must have a dot product of zero and a norm of $1$. Yet, while the dot product of these two vectors is $1$, my textbook lists these bases as orthonormal.

Answer 1

The complex inner product uses the conjugate of one of the vectors so ignoring the scalars we have that $(1,i) \cdot (1,-i) = 1 + i^2=0$ because the conjugate of $-i$ is $i$.