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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 topologicalvector spaces is illustrated by the fact that the underlying vector space ofthe Lie algebra of any pro-Lie group is weakly complete. In this study, weaklycomplete real or complex associative algebras are studied because they arenecessarily projective limits of finite dimensional algebras. The group ofunits $A^{-1}$ of a weakly complete algebra $A$ is a pro-Lie group with theassociated topological Lie algebra $A_{\rm Lie}$ of $A$ as Lie algebra and theglobally defined exponential function $\exp\colon A\to A^{-1}$ as theexponential function of $A^{-1}$. With each topological group, a weaklycomplete group algebra $\mathbb{K}[G]$ is associated functorially so that thefunctor $G\mapsto \mathbb{K}[G]$ is left adjoint to $A\mapsto A^{-1}$. Thegroup algebra $\mathbb{K}[G]$ is a weakly complete Hopf algebra. If $G$ iscompact, the $\mathbb{R}[G]$ contains $G$ as the set of grouplike elements. Thecategory of all real Hopf algebras $A$ with a compact group of grouplikeelements whose linear span is dense in $A$ is shown to be equivalent to thecategory of compact groups. The group algebra $A=\mathbb{R}[G]$ of a compactgroup $G$ contains a copy of the Lie algebra $\mathcal{L}(G)$ in $A_{\rm Lie}$;it also contains a copy of the Radon measure algebra $M(G,\mathbb{R})$. Thedual of the group algebra $\mathbb{R}[G]$ is the Hopf algebra ${\mathcalR}(G,\mathbb{R})$ of representative functions of $G$. The ratherstraightforward duality between vector spaces and weakly complete vector spacesthus becomes the basis of a duality ${\mathcal R}(G,\mathbb{R})\leftrightarrow\mathbb{R}[G]$ and thus yields a new aspect of Tannaka duality.

The Pro-Lie Group Aspect of Weakly Complete Algebras and Weakly Complete Group Hopf Algebras