Definition of Bounded Variation Function with vectorial arguments

Question

I try to found a definition of a function $$f(x)\colon\mathbb R^m\to \mathbb R^n$$ that use the norm. Is the formula below correct? $$TV=\sup\sum_{i=1}^k \|f(x_i)-f(x_{i-1})\|.$$ with k any finite integer

Answer 1

No, the definition above only works for $f: [a,b] \rightarrow \mathbb{R}^n$, in other words, where the domain is a real interval. Because what is the supremum over? On the real line, the supremum over partitions of the interval is the appropriate supremum, but over $\mathbb{R}^m$, the notion of "partition" doesn't quite work (or, if it can be generalized in a way I'm unaware of, it must be rather complicated).

To properly generalize to vector arguments, one needs to instead characterize the variation norm by weak derivatives. See Wikipedia for details.