Space of Continuous on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space

Theorem
Let $I := \closedint a b$ be a closed real interval.

Let $\map \CC I$ be the space of real-valued functions continuous on $I$.

Let $\struct {\R, +_\R, \times_\R}$ be the field of real numbers.

Let $\paren +$ be the pointwise addition of real-valued functions.

Let $\paren {\, \cdot \,}$ be the pointwise scalar multiplication of real-valued functions.

Then $\struct {\map \CC I, +, \, \cdot \,}_\R$ is a vector space.

Proof
Let $f, g, h \in \map \CC I$ such that:


 * $f, g, h : I \to \R$

Let $\lambda, \mu \in \R$.

Let $\map 0 x$ be a real-valued function such that:


 * $\map 0 x : I \to 0$.

Let us use real number addition and multiplication.

$\forall x \in I$ define pointwise addition as:


 * $\map {\paren {f + g}} x := \map f x +_\R \map g x$.

Define pointwise scalar multiplication as:


 * $\map {\paren {\lambda \cdot f}} x := \lambda \times_\R \map f x$

Let $\map {\paren {-f} } x := -\map f x$.

Closure Axiom
By Sum Rule for Continuous Real Functions, $f + g \in \map \CC I$

Commutativity Axiom
By Pointwise Addition on Real-Valued Functions is Commutative, $f + g = g + f$

Associativity Axiom
By Pointwise Addition is Associative, $\paren {f + g} + h = f + \paren {g + h}$.