= Oblique projection
{title2=$Q(v+w)=v\quad(H=V\oplus W)$}
If two <closed subspaces of a Hilbert space> give $H=V\oplus W$, the projection onto $V$ along $W$ fixes $V$ and vanishes on $W$. It need not be an <orthogonal projection>. The <bounded inverse theorem> applied to the addition map $(v,w)\mapsto v+w$ on the complete component spaces proves boundedness of the component maps. With both complementary subspaces nonzero, its <operator norm> and that of $I-Q$ equal $\sec\theta_{V,W^\perp}$, with the <directed subspace angle> convention. Indeed, for fixed $v\in V$, the smallest possible <norm> of $v+w$ over $w\in W$ is $\|P_{W^\perp}v\|$, which proves the formula for $\|Q\|$. Equality with $\|I-Q\|$ follows from the block representation $Q=\begin{pmatrix}I&B\\0&0\end{pmatrix}$ on $V\oplus V^\perp$: both squared <operator norms> are $1+\|B\|^2$. If a summand is zero, $Q$ is zero or the <identity operator>, and one complementary <operator norm> is zero; those cases must be treated directly.
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