ABSTRACT
Formal groups are cogroups in a category of a special type of topological rings. To define these rings one has to use the theory of linearly compact vector spaces. It is well known that the duality theory of vector spaces, which is a very useful tool when limited to finite-dimensional spaces, completely breaks down for infinite-dimensional spaces, the second dual of such a space being far too big and having no sensible relation to the original space. To restore the usefulness of duality theory, one has to introduce topological concepts. A way of defining the topological vector space is to say that it is the inverse limit of the discrete finite-dimensional vector spaces. It can be proved that the topological vector spaces may be characterized by the property of linear compactness: they are vector spaces with a Hausdorff topology compatible with the additive group structure, with a fundamental system of neighborhoods of 0 consisting of vector subspaces.
