The only place I've encountered well-definition is with proving an operation defined on an equivalence class is independent of the choice of representative.
On my homework, it asks us to show that the ring of formal Laurent series is well-defined, and I don't understand what exactly I need to show. However, I don't understand what I'm trying to prove. I've read the wikipedia article and don't understand how there is anything to prove in this case.
If anyone could either point me to some better references explaining well-definition or explain what I need to do to prove that a laurent series is well-defined, I would appreciate it.
Thanks, 🙂
Best Answer
I'll make my comment into an answer so that it can be marked off.
The issue is probably not whether a formal Laurent series is well-defined by itself, but rather whether the operations you are defining on formal Laurent series (and in particular the operation of multiplication of two formal Laurent series) is well-defined, in the sense that if you take any two formal Laurent series, then the definition of "product" will in fact yield a formal Laurent series. When you work with formal power series (only nonnegative exponents), one usually defines the product by: $$\left(\sum_{n=0}^{\infty}a_nx^n\right)\left(\sum_{n=0}^{\infty}b_nx^n\right) = \sum_{n=0}^{\infty}\left(\sum_{i+j=n}a_i b_j\right)x^n.$$ In this case, since $a_k=b_k=0$ if $k<0$, then the definition makes sense, as each term on the right hand side is a finite sum, which makes sense. If you try doing the same thing with formal Laurent series, where the index runs from $-\infty$ to $\infty$, then it is not obvious that this definition always results in something that you can call a formal Laurent series. So one needs to check that it does in fact yield a formal Laurent series, and that the product so defined makes the set into a ring.
So here, the issue of "well-defined"ness is not like the one when you define a function in terms of representatives (in terms of the "name" of an object when the object may have many different names), but rather in terms of whether the function actually makes sense for every input and yields an appropriate output. It is the same issue that arises if you try to define a map $f\colon(0,1)\to\mathbb{N}$ by taking a number in decimal expansion $0.a_1a_2a_3\ldots$ and "defining" $f(0.a_1a_2a_3\ldots) = \cdots a_3a_2a_1$; this map is not well-defined because for some inputs the output does not lie in the range or does not make sense.