Question 6.7.7: For f , g in C[a, b], set 〈 f,g 〉 = ∫a^b f(t)g(t ) dt Show...

Inner product Axioms 1–3 follow from elementary properties of definite integrals. For Axiom 4, observe that

\langle f, f\rangle=\int_{a}^{b}[f(t)]^{2} d t \geq 0.

The function [f(t)]^{2} \text { is continuous and nonnegative on }[a, b]. If the definite integral of [f(t)]^{2} \text { is zero, then }[f(t)]^{2} \text { must be identically zero on }[a, b], by a theorem in advanced calculus, in which case f is the zero function. Thus \langle f, f\rangle=0 implies that f is the zero function on [a, b] \text {. So }(5) \text { defines an inner product on } C[a, b] .