A resistance network is a connected graph $(G,c)$. The conductance function
$c_{xy}$ weights the edges, which are then interpreted as conductors of
possibly varying strengths. The Dirichlet energy form $\mathcal E$ produces a
Hilbert space structure (which we call the energy space ${\mathcal H}_{\mathcal
E}$) on the space of functions of finite energy.

We use the reproducing kernel $\{v_x\}$ constructed in \cite{DGG} to analyze
the effective resistance $R$, which is a natural metric for such a network. It
is known that when $(G,c)$ supports nonconstant harmonic functions of finite
energy, the effective resistance metric is not unique. The two most natural
choices for $R(x,y)$ are the ``free resistance'' $R^F$, and the ``wired
resistance'' $R^W$. We define $R^F$ and $R^W$ in terms of the functions $v_x$
(and certain projections of them). This provides a way to express $R^F$ and
$R^W$ as norms of certain operators, and explain $R^F \neq R^W$ in terms of
Neumann vs. Dirichlet boundary conditions. We show that the metric space
$(G,R^F)$ embeds isometrically into ${\mathcal H}_{\mathcal E}$, and the metric
space $(G,R^W)$ embeds isometrically into the closure of the space of finitely
supported functions; a subspace of ${\mathcal H}_{\mathcal E}$.

Typically, $R^F$ and $R^W$ are computed as limits of restrictions to finite
subnetworks. A third formulation $R^{tr}$ is given in terms of the trace of the
Dirichlet form $\mathcal E$ to finite subnetworks. A probabilistic approach
shows that in the limit, $R^{tr}$ coincides with $R^F$. This suggests a
comparison between the probabilistic interpretations of $R^F$ vs. $R^W$.