How to prove a function is vector valued

[friend]

Suppose you have a function f. Perhaps its input is one variable, f(x), or maybe its input is two variables, f(x,y). How can you prove that the function is itself vector valued,
f=(f1(x,y),f2(x,y))?

Or perhaps it would be easier to prove that f is a complex valued function,
f=f1(x,y)+if2(x,y).

 

[HallsofIvy]

All I can say is that the way it is defined tells what the "co-domain" is. I presume you are assuming that f1(x,y) and f2(x,y) are themselves real-valued and that (x, y) is a member of the vector space R². However, R², alone, is not "given" as a vector space. You would have to assume, or be given, the standard definitions for sum of two pairs and scalar multiplication.

Similarly, if f1(x,y) and f2(x,y) are real valued, then f1(x,y)+if2(x,y) is, by definition, a complex number for every (x, y). (The requirment that f1 and f2, separately, be real valued is necessary. For example, if f1(x, y)= x+ y and f2(x, y)= i, f(x,y)= f1(x,y)+ if(x,y)= x+ y- 1 which is real valued.)