# How formal or informal should math texts (written for different purposes) be?

When writing math articles (or just math text), do you write down mathematical expression in a formal way or describe it in words, e. g.

"Let $X$ be a normed vector space. Then $X$ is called a Banach space, if is complete i.e. if every Cauchy sequence in $X$ converges",

vs.

"Let $(X,+,\cdot)$ be a vector space, together with a norm $\left\Vert \cdot \right\Vert$. Then $X$ is called a Banach space $:\Leftrightarrow \ \forall x\in$ {$y \in X^\mathbb{N}\ |y \ \text{Cauchy}$}$\ \exists x_0 \in X:\ x_n \rightarrow x_0$" ?

Give me your ideas how formal do you think should a written math text be ?

Do you use different levels of formalism for different types of texts (e.g. one level for articles that are to be submitted, another one when taking notes during a lecture/working through a book and a third one, when lecturing) ?

Also, if I encounter very complicated expressions, it seems to me, that it is easier to make sense of the expression, if it is written in a formal way rather than presented in word, because it is easier to keep track of the order of the quantifiers (and thus which object depends on which) if the expression is very long (see this post of mine, where I deliberately constructed a rather long and difficult logical formula, that would be cumbersome to explain in words); do you see it like that too ?

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From a student's perspective, that second line makes me want to go read something else, even if I understand what it's saying. –  Harry Stern Nov 5 '11 at 11:25
The second version is plain wrong, as it misses the crucial assumption of $x_n$ being a Cauchy sequence. This is a perfect example of how exaggerated formalism does not add to clarity but rather obscures it, so much so, that you often don't even see that you're missing the main point... In fact, your second definition of Banach spaces excludes all non-zero spaces. –  t.b. Nov 5 '11 at 11:59
You might be interested in Paul Halmos' "How to write mathematics": dl.dropbox.com/u/1963234/halmos-howtowrite.pdf –  Gunnar Þór Magnússon Nov 5 '11 at 12:09
In words, of course. –  GEdgar Nov 5 '11 at 12:36
Using formulas or symbols instead of words doesn't make it more formal. The words 'norm', '(Cauchy) sequence', 'converges', etc. all have precise definitions; they are abbreviations if you like. –  wildildildlife Nov 5 '11 at 13:49

I started writing what follows as a comment, because I'm not really answering your question, but it wound up being way too long for a comment.

Your "formal way" is:

"Let $(X,+,\cdot)$ be a vector space, together with a norm $\left\Vert \cdot \right\Vert$. Then $X$ is called a Banach space $:\Leftrightarrow \ \forall x\in$ {$y \in X^\mathbb{N}\ |y \ \text{Cauchy}$}$\ \exists x_0 \in X:\ x_n \rightarrow x_0$"

The main problem I see with your question is that your formal way is needlessly overwritten in some ways and underwritten in other ways. For example, in writing $(X,+,\cdot)$ you introduce the symbols $+$ and $\cdot$, but then these symbols are not used later. The same goes for $\left\Vert \cdot \right\Vert$. On the other hand, $x_n$ appears later without being defined or having been previously introduced. Also, just saying that $X$ is a vector space on which a norm is defined (presumably the reader is expected to infer from the phrase "together with a norm" that the norm is actually a norm on $X$, and not a norm defined on some possibly different space) does not mean that $X$ is a normed space. Of course, this can be fixed by saying "together with a compatible norm on $X$" (assuming the reader can be expected to know what compatible means in this context), but better would be to simply say "normed space", defining "normed space" in advance if need be. Or use the phrase "normed vector space" if you wish, but off-hand I can't think of why "normed space" might be ambiguous and thus require the additional word "vector" to be used.

What I'm getting at is that simply throwing in a lot of symbols and logical notation will not make the exposition more precise. The symbols and logical notation need to be appropriate. And, if appropriate, they need to be correctly used.

Assuming "normed space" and "completeness" have been defined in advance (which they should be), the definition in words can be given as follows:

A Banach space is a normed space that is complete.

Note that I didn't need to introduce a name $X$ for the space, so I didn't. Note that I started out with the term being defined. Note that I avoided a backwards "if ... then" construction, where the hypotheses of a conditional statement isn't given until the end, such as is the case in what you wrote: "Then $X$ is called a Banach space, if [$X$] is complete". (I put square brackets around the 2nd $X$ because you didn't have a 2nd $X$ there.)

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Obviously, one would choose your first example, but not because it is "less formal". It is absolutely precise.

One uses symbols when they simplify an expression and thus make it more readable. Your use of symbols is very inconsistent, for example, you define with symbols what a sequence is, but you assume that readers know what Cauchy is.

One important ingredient of good writing is to know what you want to assume from your readers. If you are not consistent there, the question of more or less symbols is irrelevant.

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