TITLE: Show that ${1, x, x^2, x^3, ...}$ is linearly independent.
QUESTION [2 upvotes]: $Q$: Show that the non-negative powers of $x$ ($1, x, x^2, x^3,$ etc.) are linearly independent in $C^0[0,1]$ (The continuous real-valued functions on $[0,1]$) and give an example of a continuous function in $C^0[0,1]$ that is not spanned by this list.

I know one way you can prove this is by showing that if this list was linearly dependent you would end up with a polynomial with infinite roots, so it must be the constant zero polynomial, and thus this list is linearly independent.
But I'm looking for a solution using ideas from Calculus. I believe the approach is to keep taking derivatives and showing that the constants must be zero, but I'm having a hard time formalizing this into a proof for this infinite list.

REPLY [2 votes]: Suppose not, then there exists some
$$\sum_{i=0}^n a_ix^i = 0$$
(Recall that linear independence only requires taking finite sum) RHS we have the zero function. Now taking derivative $j$ times we have the constant term of the LHS to be $j!a_j$. Now evaluate at $x=0$ one observes that $j!a_j = 0$  in particular, $a_j = 0$. Since $j$ was arbitrary, it follows that all $a_j$ are zero.
So now finally, consider the exponential function $e^x$. Clearly continuous but does not lie in the span of monomials of power of $x$ since those can only form all polynomials.
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