= Solution
A form is <harmonic differential form>[harmonic] when
$$
\Delta\alpha=(d\delta+\delta d)\alpha=0.
$$
Adjointness gives
$$
\langle\Delta\alpha,\alpha\rangle_X
=\lVert d\alpha\rVert^2+\lVert\delta\alpha\rVert^2.
$$
Therefore $\Delta\alpha=0$ implies $d\alpha=0$ and $\delta\alpha=0$; the converse follows immediately from the definition of $\Delta$.
The <Hodge decomposition theorem> states in particular that every de Rham cohomology class has a unique harmonic representative. Hence
$$
\mathcal H^p(X)\xrightarrow{\sim}H^p_{\mathrm{dR}}(X),
\qquad \alpha\longmapsto[\alpha].
$$
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