polynomials, we may assume that this $p$ is homogeneous. But then $p^*(f) = p^*(f_\downarrow) \neq 0$, showing that the linear map $H \to P^*: f \mapsto (p \mapsto p^*(D)f(0))$ is 1-1; hence $P \to H^*: p \mapsto p^|_H$ is onto. Since $\dim H = \dim H_\downarrow$, and $\dim H_\downarrow = \dim P$ by assumption, the theorem follows. $\square$
We note that the converse of (4.3) Theorem does not hold, in gen- eral. For, it is easy to make up a nonhomogeneous polynomial space $H$ together with a homogeneous $P$ dual to it, for which the condi- tions $\dim(\Pi_j \cap P) = \dim(\Pi_j \cap H_\downarrow)$, all $j$, fail to hold, while these conditions are necessary for $P$ and $H_\downarrow$ to be dual, according to the following proposition of use later.
(4.4) PROPOSITION. If the homogeneous polynomial spaces Q and R are dual to each other, then
for all $j$.
Proof. Indeed, if (4.5) is violated for some (minimal) $j$ and, say, $\dim(\Pi_j \cap Q) > \dim(\Pi_j \cap R)$, then there exists a homogeneous polynomial $q \in Q$ of degree $j$ for which $q^*$ vanishes on all homogeneous polynomials in $R$ of degree $j$, and hence vanishes on all of $R$, in contradiction to the duality between $Q$ and $R$. $\square$
With this, the meaning of the following result is clear.
(4.6) RESULT [DM3]¹, [DR1]. *The polynomial spaces $\mathcal{P}(X)$ and $\mathcal{H}(X)$ are dual to each other.
In (3.12) Theorem, the space $\mathcal{P}(X)$ has been identified as the least space for certain interpolation problems. In [BR2] the space $\mathcal{H}(X)$ has been identified as the least space for other interpolation prob- lems. We now make use of the duality between $\mathcal{P}(X)$ and $\mathcal{H}(X)$ to connect $\mathcal{H}(X)$ with the interpolation problems associated with $\mathcal{P}(X)$ and vice versa. As a preparation, we procure a class of spaces- whose corresponding least space is $\mathcal{H}(X)$ in much the same way in which we obtained suitable exponential spaces $\exp_{\nu_x}$ whose least is $\mathcal{P}(X)$: We perturb the linear factors of the set of generators for the
¹The authors in [DM3] attribute the result to Hakopian.