Example of a non compactly generated complete locally convex topological vector space over $ \mathbb{R}$

Question

I am looking for an example of a non compactly generated complete locally convex topological vector space over $\mathbb{R}$. Being familiar with the fact that every complete locally convex topological vector space over $\mathbb{R}$ is a cofiltered limit of banachspaces (see https://www.math.utah.edu/~taylor/LCS.pdf Prop. 2.5) I tried to use the fact that the category of compactly generated spaces (with continuous maps as morphisms) is complete. This seems to fail because then it would be necessary that a cofiltered limit of banach spaces (in the category of topological vector spaces) is also a cofiltered limit of the underlying compactly generated spaces (in the category of compactly generated spaces). Therefore (and also because of this question https://mathoverflow.net/questions/52734/on-locally-convex-and-compactly-generated-topological-vector-spaces) I am assuming that there exists such an example but for me it is already hard enough to find a non compactly generated space besides the example on the wikipedia site. Also I need to clarify that in this context complete means that every cauchy net converges uniquely implying that the space is hausdorff.

Answer 1

After doing some research on locally convex topological vector spaces I found these two questions on Mathoverflow: https://mathoverflow.net/questions/164674/example-a-locally-convex-tvs-which-is-not-compactly-generated and https://mathoverflow.net/questions/184433/is-every-montel-locally-convex-vector-space-compactly-generate. They both seem to answer my question with the same example $\mathbb{R}^{\mathbb{R}}$ endowed with the usual product topology (or pointwise convergence). A useful paper to prove this is R. Browns Ten Topologies http://groupoids.org.uk/pdffiles/tentopologies.pdf, section 6.