= Representing functionals on harmonic forms by wedge pairing
{title2=$f(\varphi)=\int_M\varphi\wedge*h_f$}
On a compact oriented <Riemannian manifold> without boundary, every real linear functional on $\mathcal H^p$ has a unique $L^2$ representing <harmonic form> $h_f$: $f(\varphi)=\langle\varphi,h_f\rangle_{L^2}$. If $h_a$ is an orthonormal harmonic basis, then $h_f=\sum_af(h_a)h_a$. The <Hodge star> gives a harmonic $(n-p)$-form $\psi_0=*h_f$ satisfying $f(\varphi)=\int_M\varphi\wedge\psi_0$. The <ambiguity of harmonic wedge-pairing representatives> describes all smooth representatives with this same functional.
Back to article page
