= Generalized singular vector
{title2=$\|Ku_\lambda\|=1,\quad\lambda K^*Ku_\lambda\in\partial J(u_\lambda)$}
= Generalised singular vector
{synonym}
Let $K:U\to V$ be a <bounded linear operator> from a real <Banach space> to a real <Hilbert space>, and let $J$ be a proper lower-semicontinuous convex <absolutely one-homogeneous functional> on $U$. The normalization and source relation imply $\lambda=J(u_\lambda)$ by the <Euler identity for a convex one-homogeneous functional>. For the <norm> penalty $J(u)=\|u\|$ on a <Hilbert space>, its nonzero-point <subgradient> is $u/\|u\|$. Hence $K^*Ku_\lambda=u_\lambda/\|u_\lambda\|^2$: $v=u_\lambda/\|u_\lambda\|$ is a <right singular vector> with <singular value> $\sigma=1/\|u_\lambda\|$, and $Ku_\lambda$ is the corresponding <left singular vector>. Thus the definition recovers the ordinary operator singular-vector relation using the one-homogeneous norm penalty. Uniqueness of the vector from its data requires extra hypotheses on $K$ and $J$.
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