  A resistance network is a weighted graph $(G,c)$ with intrinsic (resistance)
metric $R$. We embed the resistance network into the Hilbert space ${\mathcal
H}_{\mathcal E}$ of functions of finite energy. We use the resistance metric to
study ${\mathcal H}_{\mathcal E}$, and vice versa and show that the embedded
images of the vertices $\{v_x\}$ form a reproducing kernel for this Hilbert
space. We also obtain a discrete version of the Gauss-Green formula for
resistance networks and show that resistance networks which support nonconstant
harmonic functions of finite energy have a certain type of \emph{boundary}. We
obtain an analytic boundary representation for the harmonic functions of finite
energy in a sense analogous to the Poisson or Martin boundary representations,
but with different hypotheses, and for a different class of functions. In the
process, we construct a dense space of "smooth" functions of finite energy and
obtain a Gel'fand triple for ${\mathcal H}_{\mathcal E}$. This allows us to
represent the resistance network as a system of Gaussian random variables
indexed by vertices. We also study the spectral representation for $\Delta$ on
${\mathcal H}_{\mathcal E}$ and show how nonzero defect entails a nontrivial
boundary. All of the above are are detected by the operator theory of
${\mathcal H}_{\mathcal E}$ but not $\ell^2$. Our results apply to the
Heisenberg model for the isotropic ferromagnet, improving earlier results of R.
T. Powers on the problem of long-range order (in reference to KMS states on the
$C^\ast$-algebra of the model).
