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Let $\mathcal{H}$ be a Hilbert space, $P$ and $Q$ orthogonal projections, and $\psi\in\mathcal{H}$ a unit vector. Let $R$ be the orthogonal projection onto $\overline{\text{span}(\text{im}P\cup\text{im}Q)}$. I need to show that $\langle\psi,R\psi\rangle\leq\langle\psi,(P+Q)\psi\rangle$. I feel this is extremely obvious but can't figure out how to begin the proof. Any help is much appreciated!

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This is not true. Consider $H = \mathbb{R}^2$ and let $P$ and $Q$ be the projectors on the subspaces spanned by $(1,0)$ and $(1, 1/4)$. For $\psi = (0,1)$ we can compute \begin{align} (\psi, R\,\psi) &= 1, \\ (\psi, P\,\psi) &= 0, \\ (\psi, Q\,\psi) &= 1/17. \end{align}

Or without boring calculations: let $P$ and $Q$ be the projectors on the subspaces spanned by $(1,0)$ and $(1, \varepsilon)$, $\varepsilon > 0$. For $\psi = (0,1)$ we can check \begin{align} (\psi, R\,\psi) &= 1, \\ (\psi, P\,\psi) &= 0, \\ (\psi, Q\,\psi) &< 1. \end{align}

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