Say, for instance, that I have trigonometric functions $f$, $g$, and $h$, where $f(x) = \cos(\pi x /2)$, $g(x) = \cos(\pi x/4)$ and $h(x) = \cos 4(\pi x)$
Since each function has a different period, is it correct to say they are linearly independent when they are defined for all real numbers?
If anyone is interested in the context of this question: I'm trying to show that even if a set of three functions are linearly independent when defined for all real numbers, they aren't necessarily linearly independent in the vector space of signals when we construct their singles by sampling their values at integers? I figure that I could give this counter example of functions of $f$, $g$, and $h$ that are linearly independent when defined for all real numbers, but are linearly dependent when we construct their signals by sampling their values at integers.