(a) Show that the set of all square-integrable functions is a vector space (refer to Section A.1 for the definition). Hint: The main point is to show that the sum of two square-integrable functions is itself square-integrable. Use Equation 3.7. Is the set of all normalized functions a vector space?
(b) Show that the integral in Equation 3.6 satisfies the conditions for an inner product (Section A.2).