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For planar Gaussian free field theory, the Dirichlet energy space is the completion of real test functions $C_c^{\infty }(D)$ in the norm induced by the Dirichlet inner product,On a bounded domain, the zero-boundary Poincaré inequality identifies this with $H_0^1(D)$ with an equivalent Sobolev norm. On a general unbounded domain, it must be distinguished from completion in the inhomogeneous $H^1$ norm.

A conformal bijection identifies the two corresponding energy completions by conformal invariance of the planar Dirichlet inner product. In particular, if $D$ is a nonempty simply connected proper open subset of the complex plane, the Riemann mapping theorem supplies a conformal bijection with the unit disk. This unit-disk identification is not asserted for an arbitrary multiply connected domain or the whole complex plane.

For real smooth functions of compact support in a planar domain $D$, use the Dirichlet inner productChanging the positive normalization factor does not change orthogonality. For complex functions, insert complex conjugation in the second factor to obtain the corresponding Hermitian form.

Let $f:D\to \tilde{D}$ be a conformal bijection, and let $\tilde{u},\tilde{v}$ be smooth functions of compact support on $\tilde{D}$. Its real Jacobian matrix is $Df=|f^{\prime }|R$, where $R$ is a rotation. By the chain rule,The change of variables formula has Jacobian determinant $|f^{\prime }{|}^2$, so this factor cancels:This proves conformal invariance of the planar Dirichlet inner product. By completion it is also an isometry between the corresponding Dirichlet energy spaces. It asserts invariance of the energy form, not of the inhomogeneous Sobolev norm, whose $L^2$ term has a different transformation rule.