  A bilinear form on a possibly graded vector space $V$ defines a graded
Poisson structure on its graded symmetric algebra together with a star product
quantizing it. This gives a model for the Weyl algebra in an algebraic
framework, only requiring a field of characteristic zero. When passing to
$\mathbb{R}$ or $\mathbb{C}$ one wants to add more: the convergence of the star
product should be controlled for a large completion of the symmetric algebra.
Assuming that the underlying vector space carries a locally convex topology and
the bilinear form is continuous, we establish a locally convex topology on the
Weyl algebra such that the star product becomes continuous. We show that the
completion contains many interesting functions like exponentials. The star
product is shown to converge absolutely and provides an entire deformation. We
show that the completion has an absolute Schauder basis whenever $V$ has an
absolute Schauder basis. Moreover, the Weyl algebra is nuclear iff $V$ is
nuclear. We discuss functoriality, translational symmetries, and equivalences
of the construction. As an example, we show how the Peierls bracket in
classical field theory on a globally hyperbolic spacetime can be used to obtain
a local net of Weyl algebras.
