On Bells duality theorem for harmonic functions

Define $h^∞(E)$ as the subspace of $C^∞(B̅L,E)$ consisting of all harmonic functions in B, where B is the ball in the n-dimensional Euclidean space and E is any Banach space. Consider also the space $h^{-∞}(E*)$ consisting of all harmonic E*-valued functions g such that $(1-|x|)^mf$ is bounded for some m>0. Then the dual $h^∞(E*)$ is represented by $h^{-∞}(E*)$ through $⟨f,g⟩_0= lim_{r→1}ʃ_B ⟨f(rx),g(x)⟩dx$, $f ∈ h^{-∞}(E*),g ∈ h^∞(E)$. This extends the results of S. Bell in the scalar case.