  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 required
inclusion operator is unbounded in most interesting cases. These observations
assist in the construction of the Krein extension of the Laplace operator on
$\mathcal{H}_{\mathcal{E}}$. We investigate the Krein extension and compare it
to the Friedrichs extension developed by the authors in a previous paper.
