Exercise 5.2.1. Prove Lemma 5.2.5. (Hint: the last part of (b) is a little tricky. You may need to prove by contradiction, assuming that f is not the zero function, and then show that So. If (x) |2 is strictly positive. You will need to use the fact that f, and hence If|, is continuous, to do this.) Lemma 5.2.5. Let f, g, h E C(R/Z; C). (a) (Hermitian property) We have (9, f) = (f, g). (b) (Positivity) We have (f, f) 2 0. Furthermore, we have (f, f) = 0 if and only if f = 0 (i.e., f(x) = 0 for all x E R). (c) (Linearity in the first variable) We have (f +g, h) = (f, h) + (g, h). For any complex number c, we have (cf, g) = c(f, g). (d) (Antilinearity in the second variable) We have (f, g th) = (f,g) + (f, h). For any complex number c, we have (f, cg) = c(f, g)