Exercise 1, § 4

a) There exist vector spaces E, F, G and partial operators $u,v$ from E into F, and $w$ from F into G, such that $w\circ (u+v)\not =w\circ u+w\circ v$.

b) Give an example of a closable operator $u$ on a topological vector space such that the domain of the closure of $u$ is strictly contained in the closure of the domain of $u$.

c) Give an example of closed operators $u$ and $v$ on a Hilbert space such that $u$ is continuous but $u\circ v$ is not closed.