Equivalent criteria for being dense in $L_p(X)$

Question

Is there any equivalent criteria to show that subset $A$ in the Lebesgue space $L_p(X)$ is dense? In particular, I am interested in $L_2((0,1))$.

Answer 1

Copying the comments in order to have this question answered:

We have the following proposition:

For $H=L_2(0,1)$ and a set $A$ within, consider the orthogonal of $A$, namely $A^⊥$, and the Hilbert space structure of $H$. If $A$ is dense in $H$ then $A^⊥=0.$

Moreover, if $A$ is a vector subspace of $H$ and $A^⊥=0$ then $A$ is dense in $H$.

Showing why $A$ needs to be a vector subspace of $H$ in order to have the second implication can be achieved (via contradiction) by considering the unit sphere $\mathbb{S}$ in $H$. $\mathbb{S}^⊥={0}$ yet $\mathbb{S}$ is not dense in $H$.

In order to prove the first implication, one can try the following steps: Let $A$ be dense in $H$, $a_0 \in A^⊥$. We know (why?) that there exists a sequence $(a_n)_{n\ge 1}$ of elements in $A$ such that $a_n \to a_0$.

1. Calculate the terms of the sequence $(\langle a_n, a_0 \rangle)_{n\ge 1}.$
2. Express the limit of the sequence in terms of $||a_0||$.
3. Conclude.

The full proof can be find here: If $M$ is a non-empty subset of a Hilbert space $H$, the span of $M$ is dense in $H$ iff $M^{\perp} = \{0\}$.