---
tag: 1F6SG980N5
kind: definition
name: Resistance form
msc: ["31E05", "31C25", "28A80"]
defines: THCM7DWHNY
uses: [0RBQS1JJCR, HM2HHE3J51]
sources:
  - url: https://en.wikipedia.org/wiki/Analysis_on_fractals
---

## Definition

Let $X$ be a nonempty set and let $\ell(X)$ be the vector space of all functions $X \to \R$.
Let $\mathcal{F}$ be a linear subspace of $\ell(X)$ and let $\mathcal{E} \colon \mathcal{F} \times \mathcal{F} \to \R$ be a symmetric bilinear form, with $\mathcal{E}(u) = \mathcal{E}(u, u)$.
The pair $(\mathcal{E}, \mathcal{F})$ is a *resistance form* on $X$ if:

1. $\mathcal{F}$ contains the constant functions, $\mathcal{E}(u) \ge 0$ for all $u \in \mathcal{F}$, and $\mathcal{E}(u) = 0$ if and only if $u$ is constant.
2. Let $u \sim v$ mean that $u - v$ is constant. Then $\mathcal{E}$ induces an inner product on the quotient space $\mathcal{F} / {\sim}$, and $\mathcal{F} / {\sim}$ with this inner product is a [[Hilbert space|HM2HHE3J51]].
3. For all $x \ne y$ in $X$ there is $u \in \mathcal{F}$ with $u(x) \ne u(y)$.
4. For all $x, y \in X$,
$$R(x, y) = \sup \set{ \frac{(u(x) - u(y))^2}{\mathcal{E}(u)} : u \in \mathcal{F}, \ \mathcal{E}(u) > 0 } < \infty.$$
5. (Markov property.) For every $u \in \mathcal{F}$, the function $\bar u = \min(\max(u, 0), 1)$ lies in $\mathcal{F}$ and $\mathcal{E}(\bar u) \le \mathcal{E}(u)$.

The number $R(x, y)$ is the *effective resistance* between $x$ and $y$, and $R$ is a metric on $X$, the *resistance metric*.

## Remarks

Example: let $X$ be the vertex set of a finite connected graph, with a conductance $c_{xy} = c_{yx} > 0$ on each edge $\set{x, y}$, let $\mathcal{F} = \ell(X)$, and let $\mathcal{E}(u) = \sum_{\set{x, y}} c_{xy} (u(x) - u(y))^2$, the sum over edges.
Clause 1 holds because $\mathcal{E}(u) = 0$ forces $u$ to be equal at the two ends of every edge, hence constant on the connected graph.
Clauses 2 and 4 hold because $\mathcal{F} / {\sim}$ is finite-dimensional and $\mathcal{E}$ is positive definite on it.
Clause 3 holds with the indicator function of $\set{x}$.
Clause 5 holds because $t \mapsto \min(\max(t, 0), 1)$ does not increase distances, so no term $(u(x) - u(y))^2$ grows.
Here $R(x, y)$ is the [[effective resistance|XAE5YJ3H68]] of the electrical network.

Non-example: the same formula on a graph with two connected components fails clause 1.
The function equal to $0$ on one component and $1$ on the other has energy $0$ but is not constant.

On $X = [0, 1]$, the form $\mathcal{E}(u, v) = \int_0^1 u'(s) v'(s) \, ds$ on the absolutely continuous functions $u$ with $u' \in L^2$ is a resistance form with $R(x, y) = \abs{x - y}$.
On a domain of $\R^n$ with $n \ge 2$, the Dirichlet energy $\int \abs{\nabla u}^2$ does not give a resistance form.
There are smooth functions with arbitrarily small energy that equal $1$ at a given point $x$ and vanish at another point $y$, so clause 4 fails.
The definition and the main theory of resistance forms are due to Kigami.