# Prove: every subspace of finite codimension is dense in $L^p$ with $p \in (0,1)$

$X$: 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.

• If you like the answer I suggest you upvote it and checks the "accept" check mark to the right of the answer. Otherwise some people might think you are rude. Jan 8, 2013 at 19:46

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 $L^p\to\mathbb R$.
• +1) Just a comment - the lack of continuous linear functionals on $L^p$ (where $0<p<1$) does not mean that these are useless spaces. Moreover, there are the similar looking $H^p$ (as in Hardy) spaces that do have functionals. Jan 8, 2013 at 18:39