If $X$ is a Banach space and $U,V$ are closed subspaces, such that $X \cong U \oplus V$, then a continuous linear map $P:X \rightarrow X$ is called a projection if $P|_U =id$ and ker(P)=V.
Now we can also look at the dual map $P^*:X^* \rightarrow X^*$. What is the fastest way to see that this must also be a projection? The reason I am asking this is that I want to use it in a presentation and I don't want to give lengthy proofs about the nullspace and the image, if it can be done much faster, as I am only interested in the fact that it is still a projection.
Does anybody here have a good idea how to show this?
Projections are characterized by being bounded and idempotent. The dual of $P$ is bounded and also idempotent, as can be seen by dualizing the identity $P^2=P$.