Maximum or minimum point of a vetor valued function.

Question

We know that a a point of maximum or minimum of a real-valued function can be defined. Is there some notion of maximum or minimum of a vector-valued function? For example, if $f:[0,1]\to\mathbb{R}^n$ defined by $f(t)=(f_1(t),f_2(t))$ where $f_i:[0,1]\to\mathbb{R}$ are given functions for $i=1,2$.

Thanks.

Answer 1

We talk about maximum and minimum of a real-valued function because there is a natural order in $\Bbb R$. In $\Bbb R^n$, with $n>1$, there is not. The best that you can get (again, when $n>1$) is to aim at the maximum (or minimum) of $\|f\|$ (for some norm, not necessarily the usual one).