I´m reading an introduction to $\mathbb{R}$-algebras and in the text there is an observation that says:
If $X$ is a topological space, then the set of functions $f : X \to \mathbb{R}$ are a $\mathbb{R}$ - algebra and that the continuous functions are a subalgebra of the last space.
How can I prove this?.
You just have to show that $Hom(X,\mathbb{R})$ satisfies all the conditions to be an $\mathbb{R}$-algebra. The addition of two maps $f,g$ is given by $(f+g)(x)=f(x)+g(x)$ for all $x\in X$, multiplication is given by $(f.g)(x)=f(x).g(x)$, and scalar multiplication is given by $(r.f)(x)=r.(f(x))$. From this it's easy to see that the zero function is given by $0(x)=0$ for all $x\in X$ and the unit function is given by $1(x)=1$ for all $x\in X$.
Your job is to show that this is all well-defined and satisfies the necessary axioms of an associative $\mathbb{R}$-algebra.