The anchored metric graph H1 space is a closed Hilbert subspace

The product is a Hilbert space: if a sequence is Cauchy in the finite sum-of-squares norm, each edge coordinate is Cauchy in its $H^1$ norm and converges by [F1]; the finitely many coordinate limits form an element of the product and the sum of the finitely many squared errors tends to zero. The product inner product is the sum of the edge inner products.

For each edge $e$, choose ${ϵ}_e={\mathrm{\ell }}_e\mathrm{/}2$, which satisfies the hypothesis of [F2]. For either endpoint, [F2] then gives $\max \{\mathrm{\mid }{w}_e(0){\mathrm{\mid }}^2,\mathrm{\mid }{w}_e({\mathrm{\ell }}_e){\mathrm{\mid }}^2\}\le \frac{4}{{\mathrm{\ell }}_e}\mathrm{\parallel }{w}_e{\mathrm{\parallel }}_{L^2(0,{\mathrm{\ell }}_e)}^2+{\mathrm{\ell }}_e\mathrm{\parallel }{w}_e^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^2(0,{\mathrm{\ell }}_e)}^2\le C_e\mathrm{\parallel }{w}_e{\mathrm{\parallel }}_{H^1(0,{\mathrm{\ell }}_e)}^2,$ where $C_e:=\max \{4\mathrm{/}{\mathrm{\ell }}_e,{\mathrm{\ell }}_e\}<\mathrm{\infty }$. Thus each endpoint evaluation is a bounded linear functional on its edge $H^1$ space. For every vertex, the difference of two incident endpoint evaluations is therefore continuous; its kernel is closed. Each anchor endpoint evaluation is also continuous and its zero set is closed. There are finitely many vertices, incidences, and anchors, so the intersection of these kernels, namely the trace-matching space and then $V_D$, is closed in the product.

A Cauchy sequence in either constrained space converges in the product by step 1.1. Continuity of every endpoint functional from step 1.2 passes all matching equalities and zero anchor values to the limit, so the limit remains in the same constrained space. The restricted edge-product inner product is positive definite and complete there. Hence both constrained spaces are Hilbert.