  A weakly complete vector space over $\mathbb{K}=\mathbb{R}$ or
$\mathbb{K}=\mathbb{C}$ is isomorphic to $\mathbb{K}^X$ for some set $X$
algebraically and topologically. The significance of this type of topological
vector spaces is illustrated by the fact that the underlying vector space of
the Lie algebra of any pro-Lie group is weakly complete. In this study, weakly
complete real or complex associative algebras are studied because they are
necessarily projective limits of finite dimensional algebras. The group of
units $A^{-1}$ of a weakly complete algebra $A$ is a pro-Lie group with the
associated topological Lie algebra $A_{\rm Lie}$ of $A$ as Lie algebra and the
globally defined exponential function $\exp\colon A\to A^{-1}$ as the
exponential function of $A^{-1}$. With each topological group, a weakly
complete group algebra $\mathbb{K}[G]$ is associated functorially so that the
functor $G\mapsto \mathbb{K}[G]$ is left adjoint to $A\mapsto A^{-1}$. The
group algebra $\mathbb{K}[G]$ is a weakly complete Hopf algebra. If $G$ is
compact, the $\mathbb{R}[G]$ contains $G$ as the set of grouplike elements. The
category of all real Hopf algebras $A$ with a compact group of grouplike
elements whose linear span is dense in $A$ is shown to be equivalent to the
category of compact groups. The group algebra $A=\mathbb{R}[G]$ of a compact
group $G$ contains a copy of the Lie algebra $\mathcal{L}(G)$ in $A_{\rm Lie}$;
it also contains a copy of the Radon measure algebra $M(G,\mathbb{R})$. The
dual of the group algebra $\mathbb{R}[G]$ is the Hopf algebra ${\mathcal
R}(G,\mathbb{R})$ of representative functions of $G$. The rather
straightforward duality between vector spaces and weakly complete vector spaces
thus becomes the basis of a duality ${\mathcal R}(G,\mathbb{R})\leftrightarrow
\mathbb{R}[G]$ and thus yields a new aspect of Tannaka duality.
