$X$: vector space, $N$ is subspace of $X$, the codimension of $N$ in $X$ is, by definition, the dimension of the quotient space $X/N$. Suppose $0<p<1$ and prove that every subspace of finite codimension is den in $L^p$ (Problem 1.11,p.39 in Functional analysis)(Rudin). Thanks in advance.
Suppose not, so that there is a non-dense subspace $N$ of $L^p$ of finite codimension. The closure $\bar N$ is a proper closed subspace, also of finite codimension and then $X/\bar N$ is a finite dimensional vector space of positive dimension, so there exist a non-zero linear function $\phi:X/\bar N\to\mathbb R$ and it is automatically continuous. The composition of $\phi$ with the projection map $L^p\to L^p/\bar N$ is a non-zero continuous linear map $L^p\to\mathbb R$.
This is absurd, for there are no such things.