Rudin - Principles of Mathematical Analysis, Comments and Errata

See also here for a long list from George Bergman, and see his supplement, which has lots of extra exercises. For a much shorter list of errata here.

Some hypotheses to theorems are stated a bit before the theorems rather than in the theorems themselves. So be careful.

Some theorems are not given their common name. For example Theorem 7.25 is the Arzela-Ascoli theorem.

A well known erratum is that when Rudin says X is a metric space, he almost always means that X is nonempty without stating it.

In chapter 7, problem 4, the sum doesn't make sense when $x = \frac{-1}{n^2}$. Perhaps it's part of the problem to figure that out, but it's a tad confusing. It's a fine problem to just work out for $x \geq 0$.

On problem 13 in chapter 7, in hint part (ii) Rudin constructs an f, but that will not be the f that the final subsequence converges to. In (iv) we only conclude that there is a subsequence that converges everywhere, but its limit could be different from the constructed f at countably many points. Perhaps that function should be named g or something. For extra fun, find a sequence { fn } such that the f constructed in (ii) cannot be the limit of the obtained subsequence.

On page 180, Rudin states that as E is strictly increasing and differentiable, then its inverse is also, I am not sure where he proves that. This fact requires exercise 2 on page 114 for example, in which case we obtain derivative of L directly without need for the computation afterwards. It can also be proved directly, and it is not terribly difficult but perhaps not as trivial as Rudin makes it seem.

On page 309, remark part b. The claim follows directly from the definition only if $\mu(A) < \infty$. It is slightly harder for sets of infinite measure. For example we could look at $A_j = A \cap [j,j+1)$, and find $G_j$ such that $\mu(G_j \setminus A_j) < \epsilon 2^{-|j|-2}$.

All over chapter 11, Rudin sometimes forgets to mention "measurable" for a function and then integrates it. You should think, whenever you see $\int_E f \, d\mu$, Rudin assumes f is measurable. The hypothesis of measurable is missing for example in exercises 1, 2.

In exercise 11 on page 333, and remark after Theorem 11.42 on page 329, $\mathcal{L}(\mu)$ and ${\mathcal{L}}^2(\mu)$ are metric spaces only when functions are identified if they are equal almost everywhere. So really it's the set of those equivalence classes in $\mathcal{L}(\mu)$ and ${\mathcal{L}}^2(\mu)$ that are complete metric spaces.

Some very minor errata:

• On page 213, equation (16) there is a missing vertical bar in the numerator.
• On page 313, equation (49) the sum is over i not n.