- Prove that a closed subspace of a Banach space is also a Banach space.
- Show that the linear space of all polynomials in one variable is not a Banach space in any norm.
Prove that a closed subspace of a complete metric space is complete.
The subspace of polynomials of degree $\leq n$ is closed in any norm because it is finite-dimensional. Hence the space of all polynomials can be written as countable union of closed nowhere dense sets. If there were a complete norm this would contradict the Baire category theorem.