alt Hacker News> But these excuses are not rigorously justified, even at the graduate level, in my experience.
Imo, the informal use is already pretty close to the formal definition. Formally, a distribution is defined purely by its inner products against certain smooth functions (usually the ones with compact support) which is what the OP alluded to when he said:
> The formal definition of a generalized function is: an element of the continuous dual space of a space of smooth functions.
That "element of the continuous dual space" is just a function that takes in a smooth function with compact support f, and returns what we take to be the inner product of f with our generalized function.
So (again, imo) "we don’t need to represent it directly, only the result of an inner product against a smooth function" isn't that distant to the formal definition.
I hear you, and I admit I'm drawing a fuzzy line (is the conventional approach “rigorous”).
Here are two “test functions”-
- we learned much about impulse responses, and sometimes considered responses to dipoles, etc. However, if I read the Wikipedia article correctly (it’s not great…), the theory implies that a distribution (in the technical sense) has derivatives of any order. I’m not sure I really knew that I could count on that. A rigorous treatment would have given me that assurance.
- if I understand correctly, the concept of introducing an impulse to a system that has an identity impulse response, which implies an inner product of delta with itself, is not well-defined. Again, I’m not sure if we covered that concept. (Admittedly, it’s been a long time.)