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We compare the space of square-summable functions on an infinite graph(denoted $\ell^2(G)$) with the space of functions of finite energy (denoted$\mathcal{H}_{\mathcal{E}}$). There is a notion of inclusion that allows$\ell^2(G)$ to be embedded into $\mathcal{H}_{\mathcal{E}}$, but the requiredinclusion operator is unbounded in most interesting cases. These observationsassist in the construction of the Krein extension of the Laplace operator on$\mathcal{H}_{\mathcal{E}}$. We investigate the Krein extension and compare itto the Friedrichs extension developed by the authors in a previous paper.

Unbounded containment in the energy space of a network and the Krein extension of the energy Laplacian