Basic Notions in Topological Vector Spaces

Abstract

It is only recently that theoretical physicists have become aware of the existence and possible relevance for physics of spaces other than Hilbert (or at best Banach) spaces. One even has sometimes the impression that not all of them are fully aware of the difficulties that may arise from the fact that not all Hilbert spaces are finite-dimensional. It is true that Dirac introduced and worked with δ and its derivatives long before mathematicians defined the notion of distribution, and used symbols such as the ket |p› before the notion of generalized eigenvector was made rigorous. But not everyone has the same physical intuition as Dirac had to protect him from deriving wrong conclusions by heuristic arguments, especially in the present state of physics. Of course, one cannot expect from every physicist to develop the mathematical tools he needs (though this might benefit mathematics). But one can expect from them (especially in the absence of very precise physical motivations, which seems to be the case nowadays) to know what are the existing tools, and to try and use these to make rigorous the heuristic arguments they may use as a first step.

† Presented at the NATO Summer School in Mathematical Physics, Istanbul, 1970.