Question

Prove that \(|\int_a^bf\cdot g|\le(\int_a^bf^2)^\frac12\cdot(\int_a^bg^2)^\frac12\).

Answer 1

Here again |x| is the Euclidean norm of x.

Answer 2

For this question we do have a hint: Consider separately the cases \(0 = \int_a^b(f-\lambda g)^2\) for some \(\lambda\in\mathbb{R}\) and \(0 < \int_a^b(f-\lambda g)^2\) for all \(\lambda\in\mathbb{R}\).

Answer 3

\[
< f, g > =\int_a^b f g dx
\]
is a dot product . Apply Cauchy-Schwartz Inequality
http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality
and you are done.