In V. Moretti's Spectral Theory and Quantum Mechanics, a projector on a normed space $X$ is defined as a bounded linear map $P:X\to X$ such that $P^2=P$. Is the boundedness condition really required here? I'm having trouble imagining an example of an unbounded projector. In other words, if we instead define "projector" as any linear map (on a normed space) such that $P^2=P$, do we already have that P is bounded? Does the answer change if $X$ is assumed to be a Banach space?
Note that here, we are assuming $P$ is defined on all of $X$, not only only a dense subspace (as in the linked similar question).
Let $X$ be any infinite dimensional normed linear space. Then there exists a dis-continuous linear functional $f$. Choose $x_0$ such that $f(x_0)=1$ and define $Px=f(x)x_0$. Then $P^{2}=P$ and $P$ is not bounded since $f$ is not continuous.