Required function to show that C(K) is a Vector Space. Given that K is a compact metric space and C(K) is a set of continuous real valued functions of K, what function do I use to show that C(K) is a Vector Space? I know that I have to prove that it is closed under addition and scalar multiplication but no function has been given (from my understanding). Can I just use any real valued function (f and g, for example) of K?
 A: You are correct that you need to show that it is closed under addition and scalar multiplication. This means that you need to show that if you have some compact metric space $K$, and any two functions $f,g\in C(K)$, then $f+g$ is also in $C(K)$. Similarly, for scalar multiplication you need to show that if $f$ is any continuous real-valued function and $\lambda$ is a real number, $\lambda f\in C(K)$.
There are also other axioms for a vector space that need to be satisfied for $C(K)$ to be a vector space.

*

*Associativity of addition (if $f,g,h\in C(K)$ then $(f+g)+h = f + (g+h)$

*Commutativity of addition (if $f,g \in C(K)$ then $f + g  = g + f$

*Existence of an identity (there is some $f\in C(K)$ such that for any $g\in C(K)$ $f + g = g$. We generally call $f$ 0)

*Existence of inverses (if $f\in C(K)$ then there is some $g\in C(K)$ such that $f + g = 0$. We generally call $g$ $-f$.

*Compatability of multiplication. (If $a,b \in \mathbb{R}$ and $f\in C(K)$ then $(ab)f = a(bf)$

*Identity (if $f\in C(k)$ then $1\cdot f = f$

*Distributivity. If $a,b\in \mathbb{R}$ and $f,g\in C(K)$ then $(a+b)f = af + bf$ and $a(f+g)= af + ag$.

Most of these follow relatively easily from the properties of continuity of addition and multiplication, and from the way addition and multiplication work in $\mathbb{R}$, but for $C(K)$ to be a vector space they still need to be checked.
