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In fact functions are just infinite-dimensional vectors. Almost all of the theory goes through unchanged. This is the basic idea of functional analysis.
by sverona 4y ago
In fact functions are just infinite-dimensional vectors. Almost all of the theory goes through unchanged. This is the basic idea of functional analysis.
- OJFord 4y agoOr rather size-of-their-domain-dimensional?
- sverona 4y agoThe GP kind of implied functions R->R, but yes.
- ravi-delia 4y agoIf you want, but you can do better. I believe, for instance, that at least continuous functions on the reals have a countable basis. Might even be as strong as measurable, not sure about that. That's how, for instance, fourier transforms work.
- constantcrying 4y agoThe fourier Transform essentially means that every L^2 (the square of the function has a finite integral) is completely "described" by an l^2 series (a series of numbers whose sum of squares converges), which is about the greatest piece of magic in mathematics. One very important piece here is that the term "function" is somewhat of a lie (since the result couldn't be true if it weren't). The result for measurable functions (not almost functions) shouldn't be true (I think). I am not even sure it is true for L^1 almost functions.
- constantcrying 4y agoNotably, one very important part which does not go through is that for mappings between infinite dimensional spaces linearity does not imply continuity. (E.g. a series of functions bounded by a constant can have arbitrarily large derivatives) A large part of functional analysis is dealing with that fact and its implication for PDEs.