= Kadec-Snobar projection bound
{c}
{title2=$\|P\|\le\sqrt{\dim E}$}
Every finite-dimensional <vector subspace> $E$ of a <Banach space> $X$ is the range of a bounded <linear projection> with the displayed norm bound. In maximal inscribed Euclidean or Hermitian <ellipsoid> coordinates, write the <John contact decomposition> as $\sum_rc_ru_ru_r^*=I_E$. Its contact functionals $\phi_r(x)=u_r^*x$ have norm one on $E$. Extend them to $X$ by the <Hahn-Banach theorem> and set $Px=\sum_rc_r\widetilde\phi_r(x)u_r$. This restricts to the identity on $E$. The contact synthesis operator has Euclidean norm one, the weights sum to $\dim E$, and the Euclidean unit ball is contained in the norm unit ball; these facts give $\|Px\|_E\le\sqrt{\dim E}\|x\|_X$.
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