Exercise 25

Consider the set of functions from the interval $(\,-1\ldots 1\,)\subseteq\mathbb{R}$ to $\mathbb{R}$.

• Show that this set is a vector space under the usual operations.

• Recall the formula for the sum of an infinite geometric series: $1+x+x^{2}+\cdots=1/(1-x)$ for all $x\in(-1..1)$. Why does this not express a dependence inside of the set $\{\hskip 1.00006ptg(x)=1/(1-x),f_{0}(x)=1,f_{1}(x)=x,f_{2}(x)=x^{2},\ldots\hskip 1.00006pt\}$ (in the vector space that we are considering)? (Hint. Review the definition of linear combination.)

• 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.