In this post,I will talk about the direct sum of two functions.I will divide this post into five parts.This is the first part.And below,the second and the third part will show you the definition.The fourth part will show you the proof.The last part will show you the “insights” of the direct sum of functions,so that is the most important part.
Given sets 
and their cartesian product 
.Now let’s define two maps :
and 
.The rule of these two maps are given below:
.
Given a set 
.And two functions 
,
.We define the direct sum of 
and 
as a function 
,where 
satisfies the following property:
,
. We denote as 
.
Now that has been defined,we need to prove its existence and uniqueness.I do not want to discuss the rigorous proof here.On the one hand,the rigourous proof here is easy,on the other hand, mathematics is more than rigorous proofs.Usually,long time after learning a material,what leaves in your head are not rigourous proofs,but some sort of intuitive graphs.So I’d like to talk about the direct sum intuitively(I think this intuitive proof is very near to the rigorous one):
For any fixed element 
,
is a fixed element in 
,
is a fixed element in 
.So 
is a fixed element in .So exists:
.And,because is fixed,so for any fixed 
,there is no other choice,so is unique.
Now it is the time to show the “insight” of the definition of the direct sum.(My English is not good,but I will try my best to explain it well )We know that the cartesian product consists of two ingredients: and .In fact, and are irrelevant.(By way of analogy,we can regard and as two person,they live in different places,they don’t know each other.)A map from to form a connection between and .We can see the connection between and separately,that is ,
the connection between and =the connection between and together with the connection between and .This is the spirit of direct sum.In fact, I think “direct” is a very suitable word to characterize such property.