In this paper, we prove that any perfect complex of $D^{\infty}$-modules may be
reconstructed from its holomorphic solution complex provided that we keep track of the
natural topology of this last complex. This is to be compared with the reconstruction
theorem for regular holonomic $D$-modules which follows from the well-known Riemann-Hilbert
correspondence. To obtain our result, we consider sheaves of holomorphic functions as
sheaves with values in the category of ind-Banach spaces and study some of their
homological properties. In particular, we prove that a K\"{u}nneth formula holds for them
and we compute their Poincar\'{e}-Verdier duals. As a corollary, we obtain the form of the
kernels of ``continuous'' cohomological correspondences between sheaves of holomorphic
forms. This allows us to prove a kind of holomorphic Schwartz' kernel theorem and to show
that $D^{\infty}=RHomtop(O,O)$. Our reconstruction theorem is a direct consequence of this
last isomorphism. Note that the main problem is the vanishing of the topological Ext's and
that this vanishing is a consequence of the acyclicity theorems for DFN spaces which are
established in the paper.