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  "body": "\n## Definition\n\nLet $X$ be a [[locally compact|0FE2SB8WVJ]] Hausdorff space with a countable base.\nTo each open set $U \\subseteq X$, let a linear subspace $\\mathcal{H}(U)$ of the real [[continuous functions|0X3WGX95FE]] on $U$ be given.\nIts elements are called *harmonic* on $U$.\nAn open set $V$ with compact closure $\\overline{V} \\subseteq X$ is *regular* if the following holds for every continuous $f \\colon \\partial V \\to \\R$.\nThere is a unique continuous $h \\colon \\overline{V} \\to \\R$ with $h|_V \\in \\mathcal{H}(V)$ and $h = f$ on $\\partial V$, and $h \\ge 0$ whenever $f \\ge 0$.\nThe pair $(X, \\mathcal{H})$ is a *harmonic space* if:\n\n1. $\\mathcal{H}$ is a [[sheaf|TB6TM6R4MD]]. Restrictions of harmonic functions to smaller open sets are harmonic. A continuous function on $U = \\bigcup_i U_i$ that is harmonic on each $U_i$ is harmonic on $U$.\n2. The regular sets form a base of the topology of $X$.\n3. For every open $U$ and every increasing sequence $(h_n)$ in $\\mathcal{H}(U)$ whose limit $h$ is locally bounded on $U$, the function $h$ lies in $\\mathcal{H}(U)$.\n\n## Remarks\n\nThese are the axioms in the form given by Bauer; several authors add a positivity or separation axiom, and some impose a stronger convergence axiom.\nExample: $X = \\R^n$ with $\\mathcal{H}(U)$ the classical [[harmonic functions|0DEB0JMN0W]] on $U$.\nClause 1 is clear, since harmonicity is a local condition.\nEvery ball is regular, by the Poisson integral and the maximum principle, so clause 2 holds.\nClause 3 follows from [[Harnack's principle|P15K8RKHDV]].\nA second example is $X = \\R^n \\times \\R$ with $\\mathcal{H}(U)$ the solutions of the [[heat equation|VP2HN86VAG]] $\\partial_t u = \\Delta u$ on $U$.\nNon-example: on $X = \\R^n$, let $\\mathcal{H}(U)$ be all continuous functions on $U$.\nNo open set $V$ with nonempty boundary is regular, since a continuous extension of boundary values is never unique, so clause 2 fails.\n",
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