Comparison of notation for sets Different authors use different notation, no question here...but doesn't this make the study of maths a little more difficult, always chasing different definitions of how a set is represented?  I ask this because I've studied linear algebra from two different sources; Axler and Cooperstein.  
For example, the set of all polynomials over a field $\mathbb{F}$. 
$$\text{ Axler: }\mathcal{P}(\mathbf{F})$$
$$\text{ Cooperstein: }\mathbb{F}[x]$$
I understand that, if one truly understands the material, then the notation of the set makes little to no difference.  But some notations are just a difference in font, like the real numbers; $\mathbb{R}$ or $\mathbf{R}$, simple to infer immediately.  Does there come a point where agreement is decided upon as notation or, at this level, does it really make a difference?
(I know this is a soft question, but I've never really asked and always wondered, and I suppose this is the forum for such questions, so long as the tag refers to it...)

I'm a second year graduate student and I used to wonder the same thing.  There are commonly used symbols, but by no means is there agreement on what symbol to use in every case.  I think it mainly comes down to style and the fact there are only a finite number of symbols out there to represent the ideas you want to get across.  I prefer $\mathbb{R}$ for the reals, but someone might think $\bf{R}$ looks more aesthetically pleasing (or I might already be using $\bf{R}$ to denote a ring, so I can't use it to denote the reals).  Sometimes you can't avoid introducing new notation if a symbol is already in use.  The higher up in mathematics you go, the more you start to see different notations, but it does become natural.  I hope that helps. 
