= Complementarity from two directed subspace angles
{title2=$\cos\theta_{W,V^\perp}>0,\ \cos\theta_{V^\perp,W}>0\ \Longrightarrow\ H=W\oplus V$}
For <closed subspaces of a Hilbert space>, set $T=P_{V^\perp}|_W$. The two positive cosines bound $T$ and its <adjoint operator> $P_W|_{V^\perp}$ below. The first bound gives a closed range and <injectivity>; the second makes the range dense by <image-kernel orthogonality for an adjoint>. Thus $T$ is a bounded bijection, and $w=T^{-1}P_{V^\perp}f$ yields the unique <direct sum> decomposition $f=w+v$. In equal finite dimensions, a lower bound on $T$ alone suffices by the <rank-nullity theorem>.
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