Suppose that $E = C[0,1]$ and suppose we have a metric given by $$d(f,g) = \int_0^1 \min(|f(x)-g(x)|,1)dx$$ Why is it that the topology defined by this metric makes $E$ into a topological vector space?
On the same topic, suppose that $\tau_p$ is defined by the seminorms $f \mapsto |f(x)|, x \in [0,1]$, why is the mapping $$\iota: (E, \tau_p) \rightarrow (E,d) \quad f \mapsto \iota(f)$$ is sequentially continous? To put it in other words, if a sequence in $\tau_p$ convergent it is $d$ convergent. Also, why is $\iota$ not continuous?
The topology is generated by sets of the form $$ U_{f,r}=\{g\in C[0,1]: d(\,f,g)<r\}=f+\{g\in C[0,1]: d(0,g)<r\}. $$ Note that $$ U_{0,r}+U_{0,r}\subset U_{0,2r}. $$
It suffices to show that $$ (\lambda,f)\longmapsto \lambda f,\quad\text{and}\quad (f,g)\longmapsto f+g, $$ are continuous. with respect to above topology.
a. If $f+g\in V$, $V\subset C[0,1]$ open, then $f+g\in U_{f+g,r}\subset U$, for some $r>0$, and $$ U_{f,r/2}+U_{g,r/2}\subset U_{f+g,r}\subset V, $$ and hence $(f,g)\mapsto f+g$ continuous.
b. Continuity of $(\lambda,f)\mapsto \lambda f$ is also straight-forward, starting first with the case $\lambda=0$.