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

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.
