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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 $J(u)=\Vert u\Vert$ on a Hilbert space, its nonzero-point subgradient is $u/\Vert u\Vert$. Hence $K^{\ast }K{u}_{\lambda }=u_{\lambda }/\Vert u_{\lambda }{\Vert }^2$: $v=u_{\lambda }/\Vert u_{\lambda }\Vert$ is a right singular vector with singular value $\sigma =1/\Vert u_{\lambda }\Vert$, and $K{u}_{\lambda }$ 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 $J$.

The graph norm of $A:D(A)\subseteq X\to Y$ is $\Vert x{\Vert }_A=\Vert x{\Vert }_X+\Vert Ax{\Vert }_Y$. If $X$ and $Y$ are Banach spaces, then $A$ is a closed linear operator exactly when its domain is complete in the graph norm.

If $f$ is a measurable function into a separable Banach space and has finite second moments under probability measures $\mu ,{\mu }^{\prime }$, then their Bochner integral expected values satisfyHere the Hellinger distance includes the factor $1/2$ in its square. For a proof, use $\nu =(\mu +{\mu }^{\prime })/2$, $p=d\mu /d\nu$, and $q=d{\mu }^{\prime }/d\nu$. Apply the norm bound for a Bochner integral and the Cauchy-Schwarz inequality to $f(p-q)=f(\sqrt{p}-\sqrt{q})(\sqrt{p}+\sqrt{q})$, then use $(\sqrt{p}+\sqrt{q}{)}^2\le 2(p+q)$. This explains why stability of a Bayesian posterior in Hellinger distance controls posterior expected values when the relevant second moments remain bounded.

For $1\le p<\infty$, the space ${\ell }^p$ consists of scalar sequences $x=(x_n)$ satisfyingIt is a Banach space, and ${\ell }^q\subset {\ell }^p$ when $1\le q<p$.

An operator norm limit of compact operators between Banach spaces is compact. Approximate the image of the unit ball by the image under one compact operator and transfer a finite epsilon-net across the small operator-norm error.