Finite-rank projections in a Banach space are bounded.

Question

I am curious if the following assertion is correct:

Let $X$ be a Banach space and let $P : X \to X$ be a linear map such that $P^2 = P$ and $P(X)$ is finite dimensional. Then $P$ is bounded.

I am aware of examples of unbounded operators $P$ satisfying either $P^2 = P$ or $P(X)$ being finite dimensional, but I am wondering if such operators exist when both $P^2 = P$ and $P(X)$ is finite dimensional.

Any help or comment is appreciated.

Answer 1

Let $f$ be a discontinuous linear functional on $X$, which can be constructed using a Hamel basis. Choose $x_0\in X$ such that $f(x_0)=1$, and define $$ Px = f(x)x_0. $$ Then $P$ is a discontinuous rank-one operator. $P$ is a projection because $$ P^2x = f(Px)x_0 = f( f(x)x_0)x_0 = f(x)f(x_0)x_0=f(x)x_0 = Px. $$