On locally convex (and compactly generated) topological vector spaces

Question

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.

Answer 1

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, Propositions 1.6 and 1.2), and since the cardinality of $S$ determines the isomorphism type of $\ell^{2}{(S)}$ (see here), the category of compactly generated locally convex topological vector spaces cannot be essentially small.

Answer 2

Every first countable space is compactly generated (because the topology is determined by the convergent sequences, which are the same thing as continuous maps from the compact space $\mathbb{N}\cup\{\infty\}$). Thus, if the topology on $V$ is determined by countable family of seminorms (or equivalently, it is a Fréchet space) then it is compactly generated.

Someone once told me that it is possible to develop the theory of LCTVS and duality as an application of the theory of CGWH spaces, and that this is very clean and efficient. However, I have never seen an account like this; if anyone can point me to one, I'd be very interested.

Answer 3

Since the first question doesn't seem to have been addressed directly in the answers so far, here are some suggestions. Firstly, since we are discussing topological vector spaces, I think the most natural question is to consider the forgetful function onto the category of topological spaces and since functional analysts are interested in function spaces, to completely regular spaces (of course, the case of sets can be incorporated by regarding a set as a discrete topological space). One then has a natural construction of the free locally convex space---one takes the free vector space generated by the topological space $X$ and provides it with the finest locally convex topology which agrees on $X$ with the original one. In our situation, this will be Hausdorff and will contain $X$ as a closed topological subspace. It is simple and natural to carry this one stage further and consider the completion of this space. It will have the corresonding univeral property, now for functions with values in a complete locally convex space. This space has a natural explicit representition, e.g., if we start with $[0,1]$, we get the space of Radon meassures on the interval. One of the nice things about this construction is that it can be varied almost infinitely and provides a unified approach to many spaces whose initial development was slow and painful---some of which are again forgotten lore. As examples, we can consider spaces with the universal property for bounded functions and replace continuity by other smoothness conditions---uniform continuity if $X$ is a uniform space, $C^\infty$ if $X$ is an open subset of some euclidean space, holomorphicity (subsets of the complex plane or its higher dimensional analogues), measurablility if $X$ is a measure space and so on---similarly for functions on suitable manifolds. This provides a unifying approach to such topics as uniform measures, distributions, analytic functionals and so on.

As regards the second question, I have the feeling that functional analysts and topologists use the term compactly generated with different meanings. For the former, a locally convex space (in particular, a Banach space) is compactly generated if it contains a compact subset whose span is dense. For the latter, a topological space is compactly generated if it has the finest topology which agrees with itself on compact sets (otherwise known as a $k$-space or a Kelley space). As remarked above, metric spaces have the latter property and have the former one if they are separable. Further examples of spaces which have the latter property without being metrisable are the so-called Silva spaces, i.e. countable inductive limits of sequences of Banach spaces with compact interconnecting mappings. Many of the important spaces of distributions belong to this class, as do spaces of analytic functionals.

Answer 4

Part 1: If $B$ is a basis for the vector space $X$, put the largest locally convex topology on $X$, sometimes called the direct sum topology. Trivially any mapping from $B$ into any locally convex space extends uniquely to a continuous linear mapping from $X$ into the space.

Part 2: Take a Hilbert space of any dimension but with its weak topology. Its unit ball is weakly compact.