Linear Algebra Done Right · Problem 6C.10

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · orthogonal-projections

Problem

Suppose

V

is finite-dimensional and

P \in \mathcal{L}(V)

is such that

P^2 = P

and

\|Pv\| \leq \|v\|

for every

v \in V

. Prove that there exists a subspace

U

V

such that

P = P_U