I am curious about which areas of mathematics one should be comfortable with before learning QFT. I am familiar with the "learn-it-as-you-go" approach often advocated in physics, but would like to know how to avoid that in learning QFT. Naming of specific textbooks is appreciated.
(For the sake of this discussion, "learning QFT" can be taken to mean "learning at the level of/from Peskin and Schroeder's text.")
In particular, I would like to know what, in addition to the material in Rudin's Real and Complex Analysis, one should know. Would this be sufficient? How about Folland's Real Analysis?
(I hope it's okay that I ask this question here. I've posted the same question on Physics SE but would like to hear the Math SE audience's perspective on this, especially since I imagine that more of the users on this site will have learned QFT after already knowing the relevant mathematics.)
I'm a physicist and I took a QFT course a couple of years ago and I think that, besides the analysis knowledge that you already have, you'll also need some functional analysis (specially studying Hilbert spaces) and some basic Lie theory. You should also practice the Einstein summation convention if you're not yet familiar with it.
This is off the top of my head, but I'm pretty sure that a solid knowledge in these topics should be more than enough.