*weakly complete vector spaces*as the duals of ordinary vector spaces with an emphasis on the definition of

*weakly complete topological algebras*. Their category we call $\mathcal{WA}$. We describe some basic properties of a $\mathcal{WA}$-algebra $A$ and its group $A^{-1}$ of units (i.e. invertible elements) and its exponential function. We also need the concept of a (symmetric)

*Hopf algebra*, notably within $\mathcal{WA}.$ This will allow us to define, for a topological group $G$, its (topological)

*group algebra*$\mathbb{K}[G]$ over $\mathbb{K}$. The goal is to illustrate its significance in the general theory of compact groups. As time permits we shall touch upon a parallel concept: The

*weakly complete universal enveloping algebra*$U_{\mathbb{K}}(\mathfrak{g})$ of a (profinite-dimensional) Lie algebra over $\mathbb{K}$ and the relationship to $\mathbb{K}[G]$ in view of the Lie algebra $\mathfrak{g}=\mathfrak{L}(G)$ of $G$. (This continues lectures given to the Tulane Algebra Seminar on Sep 10-2018, Mar 7-2019, Oct 2-2019; a lecture scheduled for Mar 18-2020 was cancelled due to the breakout of the Covid 19-pandemic. Main Reference: K.H.Hofmann and S.A.Morris: The Structure of Compact Groups, DeGruyter Berlin, 4th Edition 2020 [in the Tulane Library].)