# What is the Relationship between a Closed Map and a Closed Linear Operator?

A closed map between topological spaces maps closed sets to closed sets, while a closed linear operator between Banach spaces has a closed graph in the product topology.

Is this just a co-incidental use of the same "closed" terminology or is there some relationship between the two ?

• Let $$X,Y$$ be normed spaces and let $$T : X \to Y$$ be a linear operator that is neither injective nor the zero operator. Find a closed subset $$F \subset X$$ such that $$T(F)$$ is not closed in $$Y$$.
• Find a linear operator $$T : X \to Y$$ that satisfies the hypotheses of the closed graph theorem even though there is a closed subset $$F \subset X$$ such that $$T(F)$$ is not closed in the range of $$T$$.