Example of a unbounded projection Let $H$ be a Hilbert space over $\mathbb{K}$.
Let $T:H\rightarrow H$ be a linear transformation such that $T^2=T$.
What is an example of $T$ such that $T$ is unbounded?

For example: let $H = \ell^2$.  Define the transformation
$$
(x_1,x_2,\dots) \mapsto
\left(\sum_{k=1}^\infty kx_k, 0,0,\dots \right)
$$
Note, however, that this operator is not defined over all of $\ell^2$.
