Example of a unbounded projection

Question

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?

Answer 1

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 $\ell^2$.