What does the symbol $+: E\times E$ mean in $+: E\times E\to E$?

Question

What exactly does the symbol, $ +: E\times E$ mean in the formula $+: E\times E\to E$? In other word why such a symbol/operation is used to represent addition in the vector space?

Answer 1

When we write $f:A \rightarrow B$, we mean:

$f$ is a function that takes an element from set $A$ and maps it to an element in set $B$.

In the case of $+: E \times E \rightarrow E$, the function is called $+$ (that is, addition), and it takes an element from set $E \times E$ (that is, an ordered pair of elements from $E$), and maps it to an element of $E$.

Simply put, it is a function that adds two elements from $E$ to get another element in $E$.

Since the function $+$ is so common, rather than writing $+(v,u)$ to denote addtion, we usually use infix notation and write $v+u$.

---

EDIT: To clarify ordered pairs...

You may have come across the idea of an ordered pair back in high school when talking about the $xy$-plane. You know that the ordered pair $(5,-3)$ means the point where $x=5$ and $y=-3$, not $x=-3$ and $y=5$. Because the pair is ordered, you know the first element is always the $x$ coordinate and the second element is always the $y$ coordinate.

If we have the sets $A = \{a,b,c\}$ and $B=\{1,2,3\}$, then the set $A \times B$ is the set of all ordered pairs where an element of $A$ comes first and an element of $B$ comes second. For example, $(a,1)$, $(c,2)$, and $(b,3)$ are all elements of $A \times B$. Because order matters, $(3,a)$ is not an element of the set $A \times B$. Rather, it is an element of the set $B \times A$. The $\times$ symbol here is called the Cartesian product.

In the case of the set $E \times E$, the order does not play as important of a role, but you can think of the first $E$ as giving us the element on the left of the $+$ sign and the second $E$ as giving us an element on the right of the $+$ sign. So when we put the element $(v,u)$ through the function $+$, we get $v+u$.

Answer 2

The notation "$+:E\times E\to E$" is exactly the same notation as $f:X\to Y$.

The function name is $+$, the domain is $E\times E$, and the codomain is $E$. Addition in a vector space $E$ is a special example of a function between these two sets.

Answer 3

It is to symbolize that addition is an operator that takes two inputs from the vector space $E$ and then returns an element in $E$.

You may have seen the notation $$f: A\to B $$ which means that $f$ is a function that maps an element from $A$ (the domain) into $B$ (the co-domain). This is basically the same. In this example the addition operator takes two inputs (i.e. from the domain $E\times E$) and returns and element in $E$ (the co-domain).

Answer 4

It's like the definition of a function.. $$f: \mathbb R \times \mathbb C \mapsto \mathbb R$$

in this case they are telling you that the operator $+$ operates on two values taken from the same vector space $E$, that is $+: E \times E \mapsto \dots..$

And usually it maps into $E$ itself. There are other properties we usually attach to the operator $+$ when we talk about vector spaces, but that's not really relevant to your question