Re: Clarification on definition of limits

Randy Poe wrote:

Stephen J. Herschkorn wrote:


Randy Poe wrote:


Does that mean that differentiation can be defined on
a discrete space?



IIRC, a discrete space is not normalizable, so I don't think one can
come up with a useful definition of derivative.  In any case, limits are
not unique in a discrete space.

What is the point of your question?



The point of my question was that I wanted to know the
answer.



You surely already knew the answer.



Then allow me to disappoint you by my ignorance.

I was surprised by the notion that one can extend
the idea of limits and continuity to a discrete
space. So it made me wonder whether there was a way
to extend the notion of differentiability to general
spaces as well, in some way that I was not aware of.


Oh, I'm sorry. Perhaps I confused you with another poster in thinking of past posts. Really, do take a look at general topology for limits in general spaces (though in topology one usually refers to continuity rather than limits).

One *can* define derivatives on general normed spaces (over R or C): Let V and W be normed vector spaces, let A be a subset of V, let x be in A, and let f be a function from A to W. A *derivative* of f at x is a linear function L: V --> W such that |f(x) + L(y-x)- f(y)| / |y-x| vanishes as y approaches x. Note that this is not quite the same as the definition for Euclidean space, where we say the matrix of L is the derivative. See, for example, Spivak, _Calculus on Manifolds_. Note that for general normed spaces, neither existence nor uniqueness of the derivative are guaranteed.

There are other possible generalizations of differentiation, but others here know more about this than I.

--
Stephen J. Herschkorn                        sjherschko@xxxxxxxxxxxx
Math Tutor in Central New Jersey and Manhattan

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