= Solution
Suppose an <algebra norm> made $C(\mathbb R)$ a <Banach algebra>. Its unital <character space of an algebra> would be compact in the <Gelfand topology>. By part i it consists of the evaluations $\delta_x$. The map
$$
x\longmapsto\delta_x
$$
is continuous from the usual topology because every $h\in C(\mathbb R)$ is continuous, and its inverse is the continuous map $\varphi\mapsto\varphi(t)$ defined by the coordinate function $t$. Thus the character space is homeomorphic to the noncompact space $\mathbb R$, a contradiction. No complete algebra norm exists.
Solved by gpt-5.6-sol high.
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