Capitalization of named theorems I would like to extend a previous question on the capitalization of theorem names . This time, it's not about numbered theorems like "Lemma 1.1", but about theorems that carry a name.

Question: Which of the following variants is to be preferred?



*

*"By Zorn's lemma, we can conclude..."

*"By Zorn's Lemma, we can conclude..."


Additionally, I would like to know how to deal with the corner case where the name of the person after which the theorem is named starts with a lowercase letter

Question: Which of the following variants is to be preferred?



*

*"The proof of van der Waerden's theorem can be extended to ..."

*"The proof of Van der Waerden's theorem can be extended to ..."

*"The proof of van der Waerden's Theorem can be extended to ..."

*"The proof of Van der Waerden's Theorem can be extended to ..."

 A: If you are writing for a journal or book series, follow their style guide.  Both versions

Zorn's Lemma
Zorn's lemma

can be found recommended in different places.  Sometimes you could imagine more than two possibilities:

Chinese Remainder Theorem
Chinese Remainder theorem
Chinese remainder theorem

And also

Triangle Inequality
Triangle inequality
triangle inequality

A: I'd say that in general, such questions about standardization in mathematics (not only in mathematics, of course) are quite important today, because they make easier the use of search engines. So, it is relevant that we all agree to call "Zorn's lemma"  a certain result, instead of e.g. "Chain's Upper Bound Theorem", much more relevant than in the past.
In particular, spelling of names is an important issue, of course, and creates a problem, for several different transliterations from other alphabets are used: e.g., Чебышёв is romanized as Chebyshev (English), Tchebychev (French), Tschebyschow (German), Chebyshov (Spanish), Tsjebysjev (Dutch), Čebyšëv (ISO 9),...&c. Here, we should at least care to write correctly the names in their original form, which is also a point of respect (yet we meet wrong forms like Lebesque or Holder or Erdös even here at MO).
On the contrary, the issue of upper vs. lower case does not seem that relevant, as search engines are usually case-insensitive. Also, Google does not care so much about the Saxon genitive, even for queries into commas.
In conclusion, as to your questions, for your thesis I would mainy care to respect an internal coherence, and of course, the correct spelling of personal names.
