I would really appreciate it if you could give me some advice on the part (e) of Exercise 13 of first chapter of Walter Rudin's book "Functional Analysis":
Let $C$ be the vector space of all complex continuous functions on $[0, 1]$. Define \begin{equation} d(f,g) = \int_0^1 \frac{\lvert f(x) - g(x) \rvert}{1 + \lvert f(x) - g(x) \rvert} \ dx \ . \end{equation} Let $(C, \sigma)$ be $C$ with the topology induced by this metric. Let $(C, \tau)$ be the topological vector space defined by the semi-norms \begin{equation} P_x(f) = \lvert f(x) \rvert, \qquad (0 \leq x \leq 1), \end{equation}
(a) Prove that every $\tau$-bounded set in $C$ is also $\sigma$-bounded and that the identity map $id: (C, \tau) \rightarrow (C, \sigma)$ therefore carries bounded sets into bounded sets.
(b) Prove that $id: (C, \tau) \rightarrow (C, \sigma)$ is nevertheless not continuous, although it is sequentially continuous (by Lebesgue's dominated convergence theorem). Hence $(C, \tau)$ is not metrizable. Show also directly that $(C, \tau)$ has no countable local base.
(c) Prove that every continuous linear functional on $(C, \tau)$ is of the form \begin{equation} f \rightarrow \sum_{i=1}^n c_i f(x_i) \end{equation} for some choice of $x_1, \ldots, x_n$ in $[0, 1]$ and some $c_i \in \mathbb{C}$.
(d) Prove that $(C, \sigma)$ contains no convex open sets other than $\varnothing$ and $C$.
(e) Prove that $id: (C, \sigma) \rightarrow (C, \tau)$ is not continuous.
First part of this question is here, the second part is here, the third part is here and the fourth part is here. I have no idea solving this part too.
Thanks in advance.
