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Why can we reformulate $ \lvert \lvert Y-P_{[X]}Y\rvert \rvert ^{2}-\lvert \lvert Y-P_{[X_{0}]}Y\rvert \rvert ^{2}$ in the following way?
Let $[X]$ and $[X_{0}]$ denote the span of $X$ and $X_{0}$, respectively. Further, let $P_{[X_{0}]}$ denote the orthogonal projection onto the subspace $[X_{0}]$, and $P_{[X]}$ denote the orthogonal ...
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Rudin's Real and Complex Analysis, Section 9.16
In Section of 9.16 from Rudin's RCA, it says
Let $\hat{M}$ be the image of a closed translation-invariant subspace $M \subset L^2$, nder the Fourier transfrom. Let $P$ be the orthogonal projection of ...
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