Example 2

If $(E_i)_{i\in I}$ is a finite family of separated locally convex spaces and $u_i$ an element of $\mathscr{L}(E_i)$ for every $i$ in I, the endomorphism $u=\bigoplus_{i\in I}u_i$ of the locally convex space $\bigoplus_{i\in I}E_i$ is a Riesz endomorphism if and only if $u_i$ is one for every $i$.