Prove that the real vector space consisting of all continuous, real-valued functions on the interval $[0,1]$ is infinite-dimensional.
Clearly it's infinite dimensional, because if you consider say $P (\mathbb{F})$ on $[0,1]$, then there are an infinite amount of continuous real-valued functions on the interval, but how do I prove this?
Best Answer
Recall that if $V$ is a finite-dimensional vector space, then each subspace of $V$ is also finite-dimensional. So if $V$ contains an infinite-dimensional subspace, then it is infinite-dimensional. As you point out, $C[0,1]$ (the space of continuous real-valued functions on $[0,1]$) has the space $P(\mathbb{R})$ of real polynomials as a subspace. If you can show that $P(\mathbb{R})$ is not finite-dimensional, then you're done.