The following lemma can be used to show that some linear transformations are well-defined. I don't quite see that. I mean, if a linear transformation $T$ is well-defined, then if $x=y$ then $T(x)=T(y)$. Can someone explain why this lemma shows well-definedness? I even provided an example where the authors of the book claim that a linear transformation is well-defined because of that lemma. I can't figure out why. If someone could clarify this, I'd be so grateful!
"Well-defined" is not just used in the sense you gave, but can be used for any kind of situation where it is not immediately obvious that the formula that you write down for a function makes sense. In this particular case, you have an infinite sum, and for that to make sense, you must check that the sum converges. Lemma 4.3 shows that the sum converges, so the function is well-defined.