The Riesz Representation Theorem for the Dual of $L^{p}(X, \mu)$, $1 \leq p < \infty$

Prove the Riesz Representation Theorem for the case $p = 1$ by adapting the proof for the case $p > 1$.

For $1 \leq p \leq \infty$ and $q$ the conjugate of $p$, let $\left\{ f_{n}\right\} \rightarrow f$ in $L^{q}(X, \mu )$ and let $\left\{ g_{n}\right\} \rightarrow g$ in $L^{p}(X, \mu )$. Show that

Find a measure space ($X, \mathcal{M}, \mu$) for which the Riesz Representation Theorem does extend to the case $p = \infty$.

For $1 \leq p < \infty$, find a representation for the bounded, linear functionals on $\ell^{p}$.

Find a representation for the bounded, linear functionals on a Dirac measure space.

Find a representation for the bounded, linear functionals on $\mathbb {R}^{n}$.