Consider the set of functions from the interval (-1 . . . 1) ⊂ R to R.
(a) Show that this set is a vector space under the usual operations.
(b) Recall the formula for the sum of an infinite geometric series: 1 + x + x2 +. . . . . . = 1/(1 - x) for all x ∈ (-1. . . 1). Why does this not express a dependence inside of the set {g(x) = 1/(1 - x), f0(x) = 1, f1(x) = x, f2(x) = x2, . . . . } (in the vector space that we are considering)? (Review the definition of linear combination.)
(c) Show that the set in the prior item is linearly independent. This shows that some vector spaces exist with linearly independent subsets that are infinite.