Description of the Duals of Subspaces of Infinitely Differentiable Functions

We describe spaces that are dual to some subspaces of the space C^∞(D) relative to the inductive limits, where inductive limits, where \(D\subset {\mathbb{R}}^{p}\) $D\subset {\mathbb{R}}^p$ is a bounded convex domain. For any logarithmically convex space of positive numbers \(\mathcal{M}=\left\{{M}_{k},k\in {\mathbb{Z}}_{+}^{p}\right\}\) $\mathcal{M}=\left\{M_k,,,k,\in ,{\mathbb{Z}}_{+}^p\right\}$ we introduce the normed space \(C\left(D,\mathcal{M}\right)\) $C\left(D,,,\mathcal{M}\right)$ of functions \(f\in {C}^{\infty }\left(D\right)\) $f\in C^{\infty }\left(D\right)$ and prove that the Fourier-Laplace transform establishes a topological isomorphism between the strongly dual space and the projective limit of the normed spaces.