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Limits are defined for general topological spaces, even ones that don't have a metric. I'm not sure how to define a derivative without a norm though. Suppose you have a function f:REnumaElish said:IMHO another way to put the question is "can you define limits in a metric space without a norm"? If you can define limits, then you can define derivative and integral, am I not right?

^{n}-->R

^{m}It seems as though you need a way to associate a number with each displacement vector h in R

^{n}, i.e. a norm. The usual way

[tex]f'(a) = \lim_{h \rightarrow 0} \frac{f(a+h) - f(a)}{h}[/tex]

doesn't make sense because you can't divide a vector in R

^{m}by a vector in R

^{n}, or any other vector for that matter. I just looked at Spivak's "Calculus on Manifolds", and he uses a normed version

[tex]\lim_{|h| \rightarrow 0} \frac{|f(a+h) - f(a) - \lambda(h)|}{|h|} = 0[/tex]

to define the derivative of f at a as the unique linear map λ : R

^{n}-->R

^{m}that satisfies the above equation. So I guess a better way to ask my question is: Is there a way to define the derivative of a function from R

^{n}to R

^{m}that doesn't require the use of a norm?