If f and g are orthogonal, are f* and g orthogonal?

[Aziza]

I am curious:

if f and g are (complex) orthogonal functions, are f* and g also orthogonal? (* denotes complex conjugate).

I would think the answer is no, in general, but I just want to confirm

 

[Office_Shredder]

What are you integrating over and what functions are you allowed?

For example if you're integrating over the unit circle and all you ask for are integrable functions, then if f = 1/sqrt(z) and g = 1/sqrt(z) (where z has an argument between 0 and 2pi), then
\oint fg dz = 2\pi i
but
\oint f^* g dz = \oint 1 dz = 0

So f and g are orthogonal, but f^* and g are not.

 

[economicsnerd]

A simpler example for f,g\in \mathbb C^2, (which we could write as f,g: \{1,2\}\to \mathbb C if you like):

Let f:= (1,i), g:=(1,-i). Then \langle f, g \rangle = (1)(\overline{1})+(i)(\overline{-i}) = (1)(1)+ (i)(i) = 0, and \langle f^*, g \rangle = (\overline{1})(\overline{1})+(\overline{i})(\overline{-i}) = (1)(1)+ (-i)(i) = 2\neq 0, so that f,g are orthogonal, but f^*,g aren't.