Exercise 36, § 2

Let A be the algebra over $\mathbf{C}$ formed by the restrictions to $[0,1]$ of rational functions in one variable with complex coefficients. This algebra is a field.

a) We endow A with the topology of convergence in measure (INT, IV, §5, n$^o11$). Then the mappings $(x, y)\mapsto x+y$ and $(x, y)\mapsto xy$ from $A\times A$ into A are continuous, and the mapping $x\mapsto x^{-1}$ from $A^*$ into A is continuous. The topology of A is not locally convex.

b) For $n\in \mathbf{N}^*$, we define the sequence $(w_{r,n})_{r\in\mathbf{Z}}$ by $w_{r,n}= (-r+ 1)^{n(-r+1)}$ if $r <0,w_{0,n}= 1$ and $w_{r,n}= (r+ 1)^{-(r+1)/n}$ if $r >0$. If $f\in A$, we put $p_n(f) =\sum_rw_{r,n}|a_r|<+\infty$, where $\sum a_rt^r$ is the Laurent expansion of $f(t)$ at 0. Then the $p_n$ are seminorms which define on A a locally convex metrizable topology. The mapping $(x, y)\mapsto xy$ from $A\times A$ into A is continuous. The mapping $x\mapsto x^{-1}$ from $A^*$ into A is not continuous. The algebra A is not complete for this topology.