Dirac Delta Function: Definition & Mathematics

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The Dirac delta function is a mathematical construct that is rigorously defined within the framework of distributions, which falls under functional analysis. It is not merely a "fuzzy" concept but serves as a valuable tool in various mathematical and physical applications. Understanding its rigorous definition requires a solid grasp of functional analysis principles. The discussion highlights the importance of distinguishing between intuitive uses of the Dirac delta function and its formal mathematical treatment. Overall, the Dirac delta function is a well-defined entity in advanced mathematics, specifically in the context of distributions.
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It's been quite some time now since I decided to stop self-studying physics and to pay more attention to the math behind. I'm working towards gaining an understanding of 100% rigorous mathematics for now.

One thing that has always bothered me is the Dirac delta function. What I want to know is the following: is the dirac delta function rigorously defined in any area of mathematics or is it just some kind of "fuzzy" mathematics to help with computations? Secondly, if it is rigorously defined somewhere, in which topic of mathematics is it? (real analysis, complex analysis, functional analysis... ?)
 
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See distributions for the rigorous treatment. It requires knowledge of functional analysis.
 
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Thread 'About the definition of topological manifold using closed sets'

We all know the definition of n-dimensional topological manifold uses open sets and homeomorphisms onto the image as open set in ##\mathbb R^n##. It should be possible to reformulate the definition of n-dimensional topological manifold using closed sets on the manifold's topology and on ##\mathbb R^n## ? I'm positive for this. Perhaps the definition of smooth manifold would be problematic, though.
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