Hey, I often get really confused with functions question. Could someone help me with the step by step process on solving this question by choosing a value.
This question is flawed for particular reasons, and the only way to fix it is with functional analysis (way above what you’re expected to know for the GRE).
I’m curious how.
Can we remove this one, I talked to Greg about this and he said it’s not appropriate for the GRE, I was supposed to delist it, but I forgot.
That’s done.
Well, the glaring issue is that the functionals contradict each other. For example:
f(2x^2) = 2f(x^2) = 2(2x) = 4x
f(2x^2) = f((x \sqrt{2})^2) = 2 (\sqrt{2}x)^{2-1} = 2 \sqrt{2} x
From observation, if we define f to be a map from [0, \infty) to \mathbb{R} then the two requirements are only satisfied when n = 1 for which f doesn’t have an inverse (isn’t a bijection).
Thus, the natural interpretation would be to treat f as a functional on a function space assuming some sort of continuity/ linearity. In this case, f would be the derivative operator and luckily the derivative is almost bijective on C^{\infty}. It’s essentially like a fredholm operator implying that the kernel and cokernel are finite dimensional.
The good thing about fredholm operators F: X \to Y is that there is a map G: Y \to X such that F \circ G - \mathrm{id} is a compact operator (namely with finite dimensional image) and G \circ F - \mathrm{id} is also a compact operator.
Thus in the context of functional analysis, it’s basically as good as invertible.
Fair enough, thanks. I did know that this was mathematically faulty, but left it at that time because I thought it was a useful learning exercise.