Part 1:


How big is the category $TVS\_{loc.conv.}$ of locally convex topological vector spaces (and continuous maps)?


In other words (and less cheekily), is there a free locally convex TVS having any given set as basis? This would imply the functor $TVS\_{loc.conv.} \to Set$ is essentially surjective and has an adjoint.


Part 2:


Consider now the intersection $T$ of $TVS\_{loc.conv.}$ (as a subcategory of $Top$) with $CGWH$, the subcategory of $Top$ of compactly generated weak Hausdorff spaces.


How big is $T$? (Or, is $T$ essentially small?)


Note that a Banach space is locally compact iff it is finite dimensional, but I am being stupid and not remembering the relationship between local compactness and compact generation, so I can't immediately use this fact.



On locally convex (and compactly generated) topological vector spaces
Part 1: The "cheeky" answer is: **huge**. There is a left adjoint to the forgetful functor $LCTVS \to Vect$ (in particular there is a left adjoint to the forgetful functor $LCTVS \to Sets$): Equip a vector space $V$ with the locally convex topology induced by *all* linear functionals on $V$ (or as Pietro Majer put it: the topology given by all semi-norms).




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**Edit 2:**


Every linear map $f: V \to W$ is continuous: every semi-norm $|\,\cdot\,|$ on $W$ gives rise to a semi-norm on $V$ by $v \mapsto |f(v)|$. For every net $v\_{i} \to v$ we have $|f(v\_{i} - v)| \to 0$, hence $f(v\_{i}) \to f(v)$ and thus $f$ is continuous.




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**Edit:** The following summarizes what has transpired from Bill's, Neil's and my answers/comments:


Part 2: If $S$ is any set then the space $\ell^{2}(S) = \left\{\lambda = \sum\_{s \in S} \lambda\_{s} s \mid \sum\_s |\lambda\_{s}|^{2} \lt \infty \right\}$ is a Hilbert space with respect to the scalar product $\langle \lambda, \mu \rangle = \sum\_{s \in S} \lambda\_{s} \overline{\mu}\_{s}$ and it contains the free vector space on $S$. Since metrizable spaces are compactly generated and weakly Hausdorff (see [N. Strickland's notes](http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf), Propositions 1.6 and 1.2), and since the cardinality of $S$ determines the isomorphism type of $\ell^{2}{(S)}$ (see [here](http://en.wikipedia.org/wiki/Hilbert_space#Hilbert_dimension)), the category of compactly generated locally convex topological vector spaces cannot be essentially small.


