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1. A list of length n is an ordered collection of n objects, as defined by my textbook. The author notes that each list, by definition, has a finite length, so "that an object that looks like(x

_{1}, x

_{2}, x

_{3},...), which might be said to have infinite, length, is not a list."

He then defines something of the form F

^{n}to be the set of all lists of length n. Later in the chapter, when he defines vector spaces, he defines [tex]\mathbb{F}^{\infty}[/tex] to be a vector space. It's not that I have a hard time believing this, as addition and scalar multiplication are defined as expected, but I don't know how to reconcile this with what is said above. How can this [tex]\mathbb{F}^{\infty}[/tex] be defined as a vector space in terms of the concept of lists (which are by definition finite)?

2. So, for all fields, we require the numbers 0 and 1 to be distinct, because we need the additive and multiplicative identity to be different; that makes me wonder over what field is {0} a vector space?

I'm inclined to say over any field, as it seems you can only construct such a vector space from within a larger one; so {0} is a subspace of all vector spaces, but it cannot be strictly be a vector space (I mean, in the sense that it's not a subspace of some other vector space).

3. A polynomial with coefficients in F is said to be a function from F to F. My textbook notes that not all vector spaces consist of lists. This example, P(F), is given, for the reason that its elements consist of functions on F, not lists. I get that with P(F) being infinite dimensional, the concept of lists is inapplicable. But would it be accurate to say that the subspaces of P(F), bounded by degree m, are lists of functions (since lists are collections of objects, which may be functions or other abstract entities).

4. In the definition of a vector space, addition and scalar multiplication are treated as functions: "By addition on V we mean a function that assigns an element u+v belonging to V to each pair of elements u,v belong to V" and likewise for scalar multiplication. There's not much confusion here, except in what sense of function should I interpret this? Is this something trivial that should be overlooked, or is there any insight to be gained through thinking about it in terms of being a function?

Thanks in advance.