An example of an unbounded non-orthogonal projection in a Hilbert space

Question

What is an example of an unbounded non-orthogonal projection in a Hilbert spaces? Does it exist?

A non-orthogonal projection is an idempotent operator: $T^2=T$. So the question is: can such an operator be unbounded?

Answer 1

If $\phi$ is an unbounded linear functional, take $u$ so that $\phi(u) = 1$, and define $P x = \phi(x) u$.

You won't get an explicit example though, because the existence of an unbounded linear operator on an infinite-dimensional Hilbert space requires some form of the Axiom of Choice.