Many of the propositions in the author's Appendix A are of this form.

I.e., if you look at how addition on function spaces is defined pointwise, (f+g)(x) = f(x)+g(x) -- that's different meanings of (+) on either side -- that looks exactly like the defining relation of a group homomorphism, except that the symbols are backwards.

It is just unhelpful in many ways. It fixates on one particular basis and it results in a vector space with few applications and it can not explain many of the most important function vector spaces, which are of course the L^p spaces.

In most function vector spaces you encounter in mathematics, you can not say what the value of a function at a point is. They are not defined that way.

The right didactic way, in my experience, is introducing vector spaces first. Vectors are elements of vector spaces, not because they can be written in any particular basis, but because they fulfill the formal definition. And because they fullfil the formal definition they can be written in a basis.

How can an uncountably infinite set be used as an index? I was fine with natural numbers (countably infinite) being an index obv, but a real seems a stretch. I get the mathematical definition of a function, but again, this feels like we suddenly lose the plot…