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 ?