In functional analysis, it is common to endow various (infinite-dimensional) vector spaces with a variety of topologies. For instance, a normed vector space can be given the strong topology as well as the weak topology; if the vector space has a predual, it also has a weak-* topology. Similarly, spaces of operators have a number of useful topologies on them, including the operator norm topology, strong operator topology, and the weak operator topology. For function spaces, one can use topologies associated to various modes of convergence, such as uniform convergence, pointwise convergence, locally uniform convergence, or convergence in the sense of distributions. (A small minority of such modes are not topologisable, though, the most common of which is pointwise almost everywhere convergence; see Exercise 8 of this previous post).
Some of these topologies are much stronger than others (in that they contain many more open sets, or equivalently that they have many fewer convergent sequences and nets). However, even the weakest topologies used in analysis (e.g. convergence in distributions) tend to be Hausdorff, since this at least ensures the uniqueness of limits of sequences and nets, which is a fundamentally useful feature for analysis. On the other hand, some Hausdorff topologies used are “better” than others in that many more analysis tools are available for those topologies. In particular, topologies that come from Banach space norms are particularly valued, as such topologies (and their attendant norm and metric structures) grant access to many convenient additional results such as the Baire category theorem, the uniform boundedness principle, the open mapping theorem, and the closed graph theorem.
Of course, most topologies placed on a vector space will not come from Banach space norms. For instance, if one takes the space of continuous functions on that converge to zero at infinity, the topology of uniform convergence comes from a Banach space norm on this space (namely, the uniform norm ), but the topology of pointwise convergence does not; and indeed all the other usual modes of convergence one could use here (e.g. convergence, locally uniform convergence, convergence in measure, etc.) do not arise from Banach space norms.
I recently realised (while teaching a graduate class in real analysis) that the closed graph theorem provides a quick explanation for why Banach space topologies are so rare:
Proposition 1 Let be a Hausdorff topological vector space. Then, up to equivalence of norms, there is at most one norm one can place on so that is a Banach space whose topology is at least as strong as . In particular, there is at most one topology stronger than that comes from a Banach space norm.
Proof: Suppose one had two norms on such that and were both Banach spaces with topologies stronger than . Now consider the graph of the identity function from the Banach space to the Banach space . This graph is closed; indeed, if is a sequence in this graph that converged in the product topology to , then converges to in norm and hence in , and similarly converges to in norm and hence in . But limits are unique in the Hausdorff topology , so . Applying the closed graph theorem (see also previous discussions on this theorem), we see that the identity map is continuous from to ; similarly for the inverse. Thus the norms are equivalent as claimed.
By using various generalisations of the closed graph theorem, one can generalise the above proposition to Fréchet spaces, or even to F-spaces. The proposition can fail if one drops the requirement that the norms be stronger than a specified Hausdorff topology; indeed, if is infinite dimensional, one can use a Hamel basis of to construct a linear bijection on that is unbounded with respect to a given Banach space norm , and which can then be used to give an inequivalent Banach space structure on .
One can interpret Proposition 1 as follows: once one equips a vector space with some “weak” (but still Hausdorff) topology, there is a canonical choice of “strong” topology one can place on that space that is stronger than the “weak” topology but arises from a Banach space structure (or at least a Fréchet or F-space structure), provided that at least one such structure exists. In the case of function spaces, one can usually use the topology of convergence in distribution as the “weak” Hausdorff topology for this purpose, since this topology is weaker than almost all of the other topologies used in analysis. This helps justify the common practice of describing a Banach or Fréchet function space just by giving the set of functions that belong to that space (e.g. is the space of Schwartz functions on ) without bothering to specify the precise topology to serve as the “strong” topology, since it is usually understood that one is using the canonical such topology (e.g. the Fréchet space structure on given by the usual Schwartz space seminorms).
Of course, there are still some topological vector spaces which have no “strong topology” arising from a Banach space at all. Consider for instance the space of finitely supported sequences. A weak, but still Hausdorff, topology to place on this space is the topology of pointwise convergence. But there is no norm stronger than this topology that makes this space a Banach space. For, if there were, then letting be the standard basis of , the series would have to converge in , and hence pointwise, to an element of , but the only available pointwise limit for this series lies outside of . But I do not know if there is an easily checkable criterion to test whether a given vector space (equipped with a Hausdorff “weak” toplogy) can be equipped with a stronger Banach space (or Fréchet space or -space) topology.