- The central limit theorem relies on the mean and variance of your random variable existing; there are random variables for which mean and variance don't exist.
- The definition of "measure-preserving" in the ergodic theorem statement is missing the measure.
- dim (ran A)^perp = dim ker A^T can be strengthened by dropping the dim's. If v in ker A^T, then A^T v = 0. Pick any Ax in ran A; then <v, Ax> = <A^T v, x> = <0, x> = 0, so v is in (ran A)^perp.
- Others pointed out that 20 looks busted.
- The definition of Haar measure needs to fix the measure of some nonzero-measure set, or "unique" needs to be replaced with "unique up to scaling" otherwise the theorem isn't true.
- "Sounders Mac Lane"
- 25: x_0 came out of nowhere; it's unclear from the text which "invertible" is meant; the basin of convergence for Newton's method for finding a local continuation can be very small indeed.
- 36: Why is the adjoint of A named T^*?
- 46: You need some assumptions about f.