Question: Let $F([0,1], \mathbb{R})$ the space of functions of [0,1] in $\mathbb{R}$, together with the topology of the point convergence $\tau$.
Show that the set of continuous functions of [0,1] in $\mathbb{R}$ is dense in $(F([0,1], \mathbb{R}), \tau)$.

Answer: 
You can indeed use the Riesz representation theorem to solve the problem.
The hypothesis
$$ \left| \int f g \right | \leq M \| f\|\_{p}$$
shows in particular that the element $g$ defines a linear continuous functional $T\_g$ on the space $L^{p}$ via the relation
$$T\_g(f) = \int f g.$$
Now, Riesz representation theorem tells you that there exists a unique $h \in L^{q}$ that represents the functional $T\_g$ in the sense that
$$T\_g(f) = \int hf \hspace{0.9 cm} \text{for any $f \in L^{p}$.}$$
Moreover, the theorem states that $\|T\_g\| = \|h\|\_{q}$. From this you may deduce that $g=h$ and since you already have seen that $\|T\_g\| \leq M$, you inmediately obtain that $\|g \| \leq M.$
I will assume that $g$ is real valued and $p>1$. (The case $p=1$ is similar).

I will prove b) directly and a) follows from b).

Let $N$ be a positive integer and $f=(sgn \,g)|g|^{q/p}I\_{|g| \leq N}$ where $q$ is the conjugate index. Then $g$ is bounded and the given inequality becomes $\int\_{|g| \leq N} |g|^{q} \leq M (\int\_{|g| \leq N} |g|^{q})^{1/p}$. This implies that $(\int\_{|g| \leq N} |g|^{q})^{1/q} \leq M$. now let $N \to \infty$.