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I see that you added a mention of infinite dimensional vector-valued functions. That's completely beyond my expertise, but you seem to imply that the standard rules of differentiation for finite dimensional vector valued functions are not valid for infinite dimensions. Is it more complicated that just saying i goes from 1 to infinity instead of 1 to 3? MarcusMaximus (talk) 04:49, 2 July 2010 (UTC)[reply]
I now added an answer to the page. Roughly: if the number n of dimensions is finite (and we use the standard Euclidean topology), then the topology works nicely componentwise, i.e., you can compute derivatives and other limits componentwise. If n is infinite, then one usually uses a topology that is stronger, so that a limit need not exist even if a componentwise limit exists. However, you don't need these facts unless you are faced with an infinite-valued vector function, in which case you should first learn some basics of infinite-dimensional vector spaces, such as a Hilbert space. --Parodi (talk) 14:49, 1 October 2010 (UTC)[reply]
