Suggestions for writing mathematics well. Many mathematicians know little about writing well, and some, unfortunately, don't even care. The suggestions here are aimed at those who want others to be able to read their work easily. First, see "Some Hints on Mathematical Style", by David Goss (http://www.math.ohio-state.edu/~goss/hint.pdf). I have only a little to add to this--for now, just a few topics. First, Goss rightly complains about the all too common constructions "$f (x) > 0$, $x \in X$" and "$f (x) > 0$ ($x \in X$)", but I think there is an even more important reason than the one he gives for avoiding them. That is the general principle: Formulas should (almost) always be separated by words. The point is to make it visually clear on the page where formulas begin and end. Mathematics is hard enough to read that everything the author can do to help the reader is valuable. Having to look carefully for where to separate something into several formulas imposes an extra burden on the reader (even if the reader can't explicitly point to it). The examples above are corrected to read: $f (x) > 0$ for all $x \in X$ or $f (x) > 0$ for $x \in X$. Other examples: For every $x \in X$, $f (x) > 0$ is corrected to read For every $x \in X$, we have $f (x) > 0$. The phrase Let $x \in X, y \in Y$ is corrected to read Let $x \in X$ and let $y \in Y$. The phrase Let $x \in X$, $y \in Y$, and $z \in Z$ is corrected to read Let $x \in X$, let $y \in Y$, and let $z \in Z$. (Occasionally one gets really stuck, but this is rare.) Second, give explanations, definitions, reasons, etc. before they are used rather than afterwards. I have on a number of occasions read a chain of calculations, gotten stuck somewhere, and only afterwards discovered that the author is using some major theorem or some later calculation to justify the step at which I got stuck. This is extremely annoying. Thus, $A \cong B \cong C \cong D$, where the second step follows from Theorem 3.2. is corrected to Using Theorem 3.2 at the second step, we have $A \cong B \cong C \cong D$. This example is relatively mild; it is much worse if additional justification is needed at the third, fourth, and sixth steps of a calculation of seven steps spread out over a multiline display. The phrase Let $A = C (X)$, where $X$ is a compact Hausdorff space. is corrected to Let $X$ be a compact Hausdorff space. Set $A = C (X)$. Even worse is something like: Set $b = f (a)$, where $f \colon [0, 1] \to [0, 1]$ satisfies $f (0) = 0$ and f (t) = 1 for $t \geq \epsilon$, where $\epsilon > 0$ is chosen so that ... Third, give specific references. Don't write "It follows from [17] that every nuclear C*-algebra is ..."; instead, write, "It follows from Theorem 2.3 of [17] that every nuclear C*-algebra is ...". The first version is especially bad if the paper is long or if the theorem is somewhat tangential to the main results of the paper, and can be forgiven if the paper is short and the result being cited is the main result of the paper. I have seen two absolutely awful examples of this sort of thing. One was a reference simply to a paper [17] which happened to be a 100 page Memoir of the AMS; the result used was not even stated separately as a lemma but rather buried in the middle of a remark on page 61 or so. Nor did the citing paper give the statement of the result used. If I had not known ahead of time where it was, I would never ever have found it in the paper. (And the only reason I did know is that the result cited was one I had used for the same purpose in an earlier paper of mine.) The other awful example was a reference saying it followed from Section 3 of [17] that something was true. There was nothing quite like the claimed statement in Section 3 of [17]; as far as I can tell, one was supposed to modify the line of reasoning spread out over several results in Section 3 of [17] to get the claimed statement, but the author of the paper I was reading didn't want to be bothered to actually write down how to do this, preferring to leave a substantial task to every reader of his paper who wanted to figure out how he proved the claimed statement. (It would be legitimate to write something like the following: "The methods of Section 3 of [17] can be used to prove that ... . We omit the details." At least, this way, the reader knows he will have to understand of Section 3 of [17], and do some work, rather than being led to expect a theorem in Section 3 of [17] whose statement easily implies what is wanted.) Fourth, here are a few examples of grammatical mistakes and ugly phrases which one should avoid. Run-on sentences: "We have $x > 0$, therefore $x^3 > 0$." is an example of a run-on sentence: two sentences strung together with a comma. The following versions are correct: "We have $x > 0$. Therefore $x^3 > 0$." "We have $x > 0$; therefore $x^3 > 0$." A very commonly seen wrong example is: "All nuclear C*-algebras are exact, see [17]." This needs to be rewritten as "All nuclear C*-algebras are exact. See [17]." or as "All nuclear C*-algebras are exact; see [17].". (In some other languages, run-on sentences are allowed. But not in English.) "We have that": This phrase is ugly and probably grammatically wrong. Many places I see it, one should simply delete "that". Thus, "We have that $x > 0$." is corrected to read "We have $x > 0$.". Other places people want to use this construction need more rewriting to avoid. Missing "let": "Let $\varepsilon > 0$, $A$ a C*-algebra, $a \in A$" is corrected to read: "Let $\varepsilon > 0$, let $A$ be a C*-algebra, and let $a \in A$". Don't omit "let", "be", or "and". "For all $0 < a < b$": This phrase means, "for all (objects called) $0$ which satisfy $0 < a < b$", which is not what is intended. If $b$ is given, it must be "for all $a$ such that $0 < a < b$", or, here, "for all $a \in (0, b)$". If neither $a$ nor $b$ is given, then write, "for all $a$ and $b$ such that $0 < a < b$". Generally, if you have "for all" or "there is", that phrase applies to the immediate next symbol. Thus, "for all $a < b$" says, among other things, that the reader is supposed to already know what $b$ is. Another example: If $x$ is given, you can't write, "there exists $x \in U$ such that ..." or "there exists a neighborhood $x \in U$ such that ...". It must be, "there exists a neighborhood $U$ of $x$ such that ...", or (again a special situation; the analog of this doesn't always exist) "there exists a neighborhood $U \ni x$ such that ..." (with backwards "$\in$"). No hyphens with numbers. The phrases, "let $x_1, x_2, \ldots, x_n$ be $n$ points in $X$" and "let $x_1, x_2, \ldots, x_n$ be $n$ distinct points in $X$" are correct as written; adding a hyphen after "$n$" gives the wrong versions, "let $x_1, x_2, \ldots, x_n$ be $n$-points in $X$" and "let $x_1, x_2, \ldots, x_n$ be $n$-distinct points in $X$". For comparison with ordinary language, "I saw seven animals" and "I saw seven large animals" are correct, but "I saw seven-animals" and "I saw seven-large animals" are wrong.