let f and g be integrable functions on [a,b]. Show that
First, there's a (rather huge and important) mistake: it must be .
Second, you can google "Cauchy-Schwartz inequality" , or: let , and then
, and now take , so substituting above:
And from here your inequality follows at once ( note that we must assume something to justify the above proof: what is this and what happens if the assumption isn't fulfilled?)