Solve the problem.For f, g in C[a, b], set = .Show that defines an inner product of C[a, b].

Question

What will be an ideal response?

Answer 1

Answers will vary.
Inner product Axioms 1-3 follow from elementary properties of definite integrals.  
Axiom 4:   = dt ? 0.
The function [f(t)]² is continuous and nonnegative on [a, b]. If the definite integral of [f(t)]² is zero, then [f(t)]² must be identically zero on [a, b]. Thus,  = 0 implies that f is the zero function on