I'm reading Serge Lang's (S.L) linear algebra book. In the beginning, at function spaces section there is such a text:
Let $S$ be a set and $K$ a field. By a function of $S$ into $K$ we shall mean an association which to each element of $S$ associates a unique element of $K$. Thus if f is a function of $S$ into $K$, we express this by the symbols
$$f:S \rightarrow K$$
We also say that $f$ is a $K$-valued function. Let $V$ be the set of all functions of $S$ into $K$. If $f$, $g$ are two such functions, then we can form their sum $f$ + $g$. It is the function whose value at an element $x$ of $S$ is:
$$f(x) + g(x)$$
We write
$$(f + g)(x) = f(x) + g(x)$$
If $c \in K$, then we define $cf$ to be the function such that
$$(cf)(x) = cf(x)$$
Thus the value of $cf$ at $x$ is $cf(x)$. It is then a very easy matter to verify that $V$ is a vector space over $K$. We shall leave this to the reader.
From what I know, set $V$ is a vector space over field $K$ iff it has:
- Associative property of addition
- Additive inverse of every element equal to additive identity ($0$)
- Commutative property of addition
- Distributive property for arbitrary scalar multiplied by sum of its elements
- Distributive property for element multiplied by sum of arbitrary scalars
- Multiplicative associative property
- Unaffected elements when multiplied by multiplicative identity ($1$)
- Every linear combination of elements belonging to the set
From what I can see, N4 is satisfied, but how can I prove that function space satisfies other properties as well? i.e commutative property