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I have never seen these index functions used as a transfinite basis for a vector space. And it seems like the function is not a limit point of finite sequences
by lanstin 3y ago
I have never seen these index functions used as a transfinite basis for a vector space. And it seems like the function is not a limit point of finite sequences of basis functions, but some weird transfinite sum with mostly zero entries? Clearly there is no Fourier transformation possible on all functions? I think diagonalization methods would be easy to disprove any useful result.
Even Hilbert spaces are usually just indexed by the ints. And such a basis gives you zero continuity or differential conditions. All the functional analysis I have seen uses some continuity conditions and has some countable basis. Other than that, it is a very useful perspective on functions, and kind of the required start to understanding quantum mechanics formalism.
- xigoi 3y agoMathematicians HATE this weird trick! Learn how he constructed a basis for the vector space of real functions without the axiom of choice.
- movpasd 3y agoThis is my major gripe with the article. While useful for intuition, that "..." after the sum is mathematically meaningless. This is also a common issue with quantum mechanics as taught at an introductory level. But it seems the article, similarly to intro QM courses, is more about motivating functional analysis concepts, which is useful as exposition even if not rigorous.
- rrobukef 3y agoThe article is some summary of a book with chapters. At some point they limit the space to the subspace of functions periodic over (b-a) and change the basis (with proof) from dirac delta to sines of frequency 2pi*k/(b-a) [with k in N]. In this subspace all functions have Fourier transformations.
- lanstin 3y agoBig changes there.
- constantcrying 3y agoFor something to be a basis each element needs to be the finite linear combination. Of course there are notions for different kinds of choices of basis, where you have countable linear combinations. The article (maybe for good reason) completely ignores the (usually quite hard) choice in functional analysis which vectorspace to use. And the one outlined here, where functions are defined pointwise, is amost always the most useless one to consider. Although I get that it was done to teach the general outline of the subject, which is quite valuable on its own. >Clearly there is no Fourier transformation possible on all functions? You don't even get a useful metric for such a space.