An infinite dimensional closed subspace of $C[0,1]$

Question

I've been studying Functional Analysis for a test and this was one of the questions in one of the previous tests:

Construct an infinite-dimensional closed subspace of $C[0,1]$ with respect to the supremum norm.

I know every infinite-dimensional Banach Space has a subspace which is not closed but I cannot find a closed one. Also, why does a supremum norm matter?

Answer 1

As mentioned earlier, $C([0,1])$ is of course a closed subspace of itself and you can verify that it has infinite dimension. Regardless, if you meant to ask for a proper subspace, then $$ C_0([0,1]) = \{ f\in C([0,1]): f(0)=f(1)=0\} $$ works. I'll leave it to you to show why.

Answer 2

We can use some abstract machinery to generalize to any infinite-dimensional normed space $X$. Pick some $0\neq v\in X$ and define the linear function $f: \mathbb{R} v \rightarrow \mathbb{R}, \ rv \mapsto r$. By the Hahn-Banach Theorem there exists a continuous linear function $\varphi: X\rightarrow \mathbb{R}$, such that $\varphi(rv)=r$. Thus, $Ker(\varphi)$ is a closed infinite-dimensional subspace of $X$. As $\varphi$ is not identically zero we get that $Ker (\varphi)$ is also proper.