# Math Help - Space of polynomials in C0

1. ## Space of polynomials in C0

Can someone show me how the space of polynomials which take [a,b] to R are a "thin" subset of C0[a,b], i.e. the space of continuous functions from [a,b] to R.

Note: A "thin" subset is sometimes also called a "meager" subset and basically means that P can be written as the countable union of closed and nowhere dense sets:

$P = \bigcup_{k=1}^{\infty}H_k$ where $H_k$ is closed and nowhere dense.

2. Consider $H_k$ the subspace of polynomials of degree smaller or equal than k. This is closed, because it's a finite dimensional subspace, and has empty interior, because a proper subspace always has empty interior. The last assertion can be proved by showing that assuming the contrary would imply the existence of a ball centered at zero included in the subspace, a contradiction.