Let (mathcal{D} subset L^{2}(mu)) and write (Sigma:=operatorname{span}(mathcal{D})). Show that (Sigma) is dense if, and only if (mathcal{D})

Question:

Let $\mathcal{D} \subset L^{2}(\mu)$ and write $\Sigma:=\operatorname{span}(\mathcal{D})$. Show that $\Sigma$ is dense if, and only if $\mathcal{D}$ is total, i.e. \[\bar{\Sigma}=L^{2}(\mu) \Longleftrightarrow\left[\forall \phi \in \mathcal{D}:\langle u, \phiangle_{L^{2}(\mu)}=0 \Longrightarrow u=0ight.\]

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