Why do we generalize functions by functionals on Schwartz Spaces, beyond the fact that it simply works? There should be a deeper reason why Schwartz considered functionals. Excited for answers, Alex.
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1$\begingroup$ I voted to close because it seems to discussiony, but maybe I was mistaken to do that. There could be some really good answers to this question. $\endgroup$– Steven GubkinCommented Jan 14, 2014 at 0:22
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1$\begingroup$ I think fundamentally, it comes down to the philosophy embodied by the yoneda lemma. To understand "blah" look at its relationship to all other things, or maybe just one particular thing. Often such a representable functor captures ALL data about a thing. Then generalized blahs come from functors which were not representable to start with. In other words, studying the category of presheaves on a given category is almost always worthwhile. $\endgroup$– Steven GubkinCommented Jan 14, 2014 at 0:25
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$\begingroup$ ...hmm ...and what does it meanin this case? $\endgroup$– C-star-W-starCommented Jan 14, 2014 at 0:48
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$\begingroup$ These objects appeared naturally long before Schwartz made them rigorous. See the history section of en.wikipedia.org/wiki/Dirac_Delta_function for motivation. There are other generalizations, e.g., hyperfunctions, so Schwartz's solution is not unique by any means. $\endgroup$– S. Carnahan ♦Commented Jan 14, 2014 at 0:52
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4$\begingroup$ Although closed: There is a fundamental physical motivation for distributions which is nearly explained in Strichartz book "A Guide to Distribution Theory and Fourier Transforms", chapter 1. In a nutshell: If $f$ shall represent a physical quantity (like temparature, pressure,…) then it does not seem plausible from a physical point of view to talk about values of $f$ and some point $x$ because true point measurements are not possible. However, averaged measurements are possible and this gives rise to testing the quantity against functions and there you are. $\endgroup$– DirkCommented Jan 14, 2014 at 8:05
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