Linear indepedent functions and complex conjugation

[jdstokes]

Suppose \{\varphi_i\} is an infinite set of linearly independent functions. Is \{ \varphi_i^\ast \} linearly indepedent? How about \{ \varphi_i \} \cup \{ \varphi_i^\ast\}?

[morphism]

Well, what are your thoughts on the matter?

[jdstokes]

Well, if \{ \varphi_i \} is LI then \{ \varphi_i^\ast \} is also trivially LI, because if it wasn't then you could just take the complex conjugate violating linear indepedence of \{ \varphi_i \}.

It's not clear if my second claim is true, although I'd like it to be, I suspect there are counterexamples waiting to be found. I'd like to be proven wrong, however.

[morphism]

What if we take, say, the set \{i f, f\}, where f is some real-valued function?