= Orthogonal direct sum
{title2=$V\oplus^\perp W$}
An orthogonal direct sum is a <direct sum> whose distinct summands are mutually <orthogonal>. For $v\in V$ and $w\in W$, the <Pythagorean identity> gives $\|v+w\|^2=\|v\|^2+\|w\|^2$. In a <Hilbert space>, a <closed subspace of a Hilbert space> and its <orthogonal complement> give an orthogonal direct sum of the entire space. The component maps are <orthogonal projections> and have <operator norm> one when their ranges are nonzero.
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