On Bell's duality theorem for harmonic functions

Define ${\displaystyle h^{\infty }\left(E\right)}$ as the subspace of ${\displaystyle C^{\infty }(B̅L,E)}$ 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 ${\displaystyle h^{-\infty }(E\cdot )}$ consisting of all harmonic E*-valued functions g such that ${\displaystyle {(1-\left|x\right|)}^{m}f}$ is bounded for some m>0. Then the dual ${\displaystyle h^{\infty }(E\cdot )}$ is represented by ${\displaystyle h^{-\infty }(E\cdot )}$ through ${\displaystyle ⟨f,g{⟩}_0=\underset{r\to 1}{lim}{ʃ}_B⟨f\left(rx\right),g\left(x\right)⟩dx}$, ${\displaystyle f\in h^{-\infty }(E\cdot ),g\in h^{\infty }\left(E\right)}$. This extends the results of S. Bell in the scalar case.