Let $\mathbb{E}$ be a (real or complex) Banach space (complete normed space). Let $P$ be a projection ($P^2=P$) from $\mathbb{E}$ into itself.
Is it necessarily that the norm of $P$ equals to $1$?
2026-03-27 16:20:06.1774628406
Projections on Normed Spaces
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This is not true in general (e.g. with $P=0$). However, if the projection is orthogonal and nonzero, then the norm is certainly $1$. See, for instance, Operator norm of orthogonal projection.