Originally posted by royalchicken
Following on something I just read:
Here's one that's a total triviality if you know what an inner product space is and quite difficult if not:
If f:[0,pi] --> R is continuous then the square of the integral of f(x)sin(x) over the domain of f is at most the integral of f^2(x)sin(x) over the same interval, times some constant. Find the smallest ...[text shortened]... is hard but not impossible to do straight out, without the abstraction which makes it very easy.
Hey royalchicken, I tried looking up "inner product space" on mathworld.wolfram.com, and it said the following:
"A vector space together with an inner product on it is called an inner product space."
It also gave an example which seemed pertinent:
"3. The vector space of real functions whose domain is an closed interval [a,b] with inner product:
= int[f*g*dx] (a|b)"
It seems to me that this is saying if you have two functions f and g, and their inner product (calculated the same way as the dot product = x^2 + f*g?) is equal to the above integral when evaluated from a to b, then you have an inner product space. But not any f(x) will produce an inner product space in the question as written, will it?
I need a hint.