## Real and Complex Projections

A projection $P$ on a real vector space is defined to be a linear mapping such that $P^2 = P$. For projections on complex vector spaces why does one require the extra condition that $P^* = P$, where $P*$ is the Hermitian of $P$?

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You're getting your concepts confused. $P^2=P$ defines projections in any vector space; the additional constraint $P^*=P$ only makes sense in inner product spaces and defines orthogonal projections. (Projections don't have to be orthogonal.)