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1. Summation. Let X be a set.
 

Summary: 1. Summation.
Let X be a set.
1.1. Finite summation. Suppose Y is a set and
+ : Y Y Y
is such that
(i) x + (y + z) = (x + y) + z whenever x, y, z Y ;
(ii) x + y = y + x whenever x, y Y ;
(iii) there is 0 Y such that y + 0 = y = 0 + y whenever y Y .
For example, Y could be an Abelian group or Y could be [0, ] where + on
[0, ) [0, ) is addition in the Abelian group of R and where
y + = = + y whenever y [0, ].
Definition 1.1. For f, g Y X
we define f + g Y X
by letting
(f + g)(x) = f(x) + g(x) for x X
and we note that appropriately reformulated versions of (i),(ii) and (iii) hold. We
let
0 : X Y
be such that 0(x) = 0 for x X.
Definition 1.2. For f Y X

  

Source: Allard, William K. - Department of Mathematics, Duke University

 

Collections: Mathematics