Since all norms on $\mathbb R^n$ are equivalent, the following question makes sense:
Can we define the notion of "differentiability" of a map $\mathbb R^n \to \mathbb R$ without refering to a norm at all? Can we define the derivative itself?
(That is, without mentioning any kind of norm, or a distance induced by it).
In fact, I guess that one could ask that even for $n=1$.
from Hot Weekly Questions - Mathematics Stack Exchange
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