Finding a norm 1+epsilon projection onto a subspace of Lp

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Hey guys I have a quick question. Let $\mu$ be a probability measure and let $E$ be a closed subspace of $L_p(\mu)$, $1\leq p\leq \infty$ which has either finite dimension or codimension. For any $\epsilon>0$, is it always possible to find a linear projection onto $E$ of norm $<1+\epsilon$?

Thanks!