22 Oct '05 22:031 edit

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 constant for which this inequality holds for all such f.

There are all sorts of similar inequalities, but I chose this one because it is hard but not impossible to do straight out, without the abstraction which makes it very easy.

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 constant for which this inequality holds for all such f.

There are all sorts of similar inequalities, but I chose this one because it is hard but not impossible to do straight out, without the abstraction which makes it very easy.