If we consider projection of $\Bbb R\times \Bbb R$ onto the $x$-axis(codomain is $\Bbb R\times \Bbb R$) the map is not closed as we know the image of a hyperbola ($xy=1$, closed) is not closed.
In my functional analysis book
Theorem : Let $X$ be a normed space and $P$ is a projection form $X$ to $X$. Then $P$ is closed map if and only if the subspaces Range $R(P)$ and Zero space $Z(P)$ are closed in X.
Definition of projection given is $P\circ P=P$.
the above example satisfies all the conditions in Theorem but the map is not closed.
please help me with this.
The definition of a close map in Functional Analysis is different from the definition in general topology. In FA a linear map is closed iff its graph is closed, but in topology a map is closed iff it maps closed sets to closed sets.