# Here are some phrases typical of the crustacean style

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Here are some phrases typical of the crustacean style: ‘The key point of the proof is to show that f is a well defined isomor- phism.’ ‘We first show that f is well defined.’ ‘Next we show that f is a morphism.’ ‘Since f is clearly surjective we need only check that f is injective which we do by looking at the kernel.’ ‘This completes the proof that f is a well defined isomorphism.’ ‘The next three lemmas are entirely routine and show that our results on continuous functions can be extended to distributions.’ ‘Lemma 3 can be improved to show that the growth is no faster than polynomial but we only need some bound depending on n alone.’ ‘This is the only point in the argument where we use Axiom A.’ Often it is better to say ‘By Theorem 7.3 which says that all snarks are boojums’ than ‘By Theorem 7.3’ or ‘Since all snarks are boojums’. In the same way ‘We show that G is Abelian’ may be less helpful to the reader than ‘We show that the group G of translations is Abelian’ or ‘We show that the group G defined at the start of Section 2 is Abelian’. Not all readers have perfect memories. 4.4 Your outline Once you have decided what your subject is you must decide how to present it. Which points should you make in the introduction? Which definitions will you need and where should they be put? What notation are you going to use? (If you use i for the identity map and go on to talk about complex 12

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numbers you, or at least your readers, may have problems.) Which lemmas will you need to prove the central theorem and in which order should they come? Should the counter-examples be presented early to show how strong the main theorem is or late to show which avenues for generalisation are blocked? The standard advice with which I have no reason to disagree says that you should start by writing down a paragraph in the style of an undergraduate syllabus. Inversion theorems of classical Fourier Analysis for R and T . Definition of a Locally Compact Abelian Group. Statement (without proof) of existence of Haar measure. Definition of char- acter. Inversion theorem corresponds to existence of ‘sufficiently many’ characters. Proof (follow Rudin) of inversion theorem for LCA group (giving parallels with classical case). Statement struc- ture theorem and brief sketch proof. Next write out the statements of your main definitions, examples, lemmas and theorems in the order that you intend to give them. You have now decided your strategy leaving your tactics (the proofs and the connecting discussions) for the first draft. At this point you should consult the assessor to check that your plans are reasonable. Do remember that the logical order is not necessarily the pedagogic order.
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