Approximation and Separability

Let $A \subseteq B \subseteq X$. Show that $A$ is dense in $B$ if and only if each $f \in B$ is the limit of a sequence $\left\{ f_{n}\right\}$ in $A$.

Let $\mathcal{F}$ be a dense subset of $X$ and suppose that $\mathcal{S} \subseteq X$ has the property that each $f \in \mathcal{F}$ is the limit of a sequence $\left\{ f_{n}\right\}$ in $\mathcal{S}$. Show that $\mathcal{S}$ is dense in $X$. In particular, show that if $\mathcal{S}$ is countable, then $X$ is separable.

Show that, for $1 \leq p < \infty , \ell^{p}$ is separable, but $\ell^{\infty }$ is not separable.

Show that the subspace of simple functions is dense in $L^{\infty }(E)$.

Show that $C[a, b]$ is not a dense subspace of $L^{\infty }[a, b]$.

Let $1 \leq p < \infty , q$ be the conjugate of $p$, and $\mathcal{S}$ be a dense subset of $L^{q}(E)$. Show that if $g \in L^{p}(E)$ and $\int_{E} f \cdot g d m = 0$ for all $f \in \mathcal{S}$, then $g = 0$.

Exhibit a measurable set $E$ for which $L^{\infty }(E)$ is separable. Show that $L^{\infty }(E)$ is not separable if the set $E$ contains a non-degenerate interval.