Can each finite rank projection in $B(H)$ be written as a finite sum of rank one projections ?Who can give me a concise proof?Thanks.
2026-03-30 02:06:47.1774836407
decomposition of finite rank projection
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Sure. Let $V$ be the (finite-dimensional) subspace you project to and denote your projection by $P_V$. Further let $\{v_1,\ldots,v_n\}$ be any basis of V and let $Q_j$ be the projection inside $V$ onto $\operatorname{span}\{v_j\}$ along all the other $v_i$'s. Then $$ P_V = (Q_1+\ldots+Q_n)P_V = Q_1P_V + \ldots + Q_nP_V, $$ which is a sum of one-dimensional projections.