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 Functions, $f + g \in \map \CC I$ {{qed|lemma}} === Commutativity Axiom === By Pointwise Addition on Real-Valued Functions is Commutative, $f + g = g + f$ {{qed|lemma}} === Associativity Axiom === By Pointwise Addition is Associative, $\paren {f + g} + h = f + \paren {g + h}$. {{qed|lemma}} === Identity Axiom === {{begin-eqn}} {{eqn | l = \map {\paren {0 + f} } x       | r = \map 0 x +_\R \map f x       | c = {{Defof|Pointwise Addition of Real-Valued Functions}} }} {{eqn | r = 0 +_\R \map f x       | c = Definition of $\map 0 x$ }} {{eqn | r = \map f x }} {{end-eqn}} {{qed|lemma}} === Inverse Axiom === {{begin-eqn}} {{eqn | l =  \map {\paren {f + \paren {-f} } } x        | r = \map f x +_\R \map {\paren {-f} } x       | c = {{Defof|Pointwise Addition of Real-Valued Functions}} }} {{eqn | r = \map f x +_\R \paren {-1} \times_\R \map f x        | c = Definition of $\map {\paren {-f} } x$ }} {{eqn | r = 0 }} {{end-eqn}} {{qed|lemma}} === Distributivity over Scalar Addition === {{begin-eqn}} {{eqn | l = \map {\paren { \paren {\lambda +_\R \mu} f} } x       | r = \paren {\lambda +_\R \mu} \times_\R \map f x       | c = {{Defof|Pointwise Scalar Multiplication of Real-Valued Functions}} }} {{eqn | r = \lambda \times_\R \map f x +_\R \mu \times_\R \map f x       | c = Real Multiplication Distributes over Addition }} {{eqn | r = \map {\paren {\lambda \cdot f} } x +_\R \map {\paren {\mu\cdot f} } x       | c = {{Defof|Pointwise Scalar Multiplication of Real-Valued Functions}} }} {{eqn | r = \map {\paren {\lambda \cdot f + \mu \cdot f} } x       | c = {{Defof|Pointwise Addition of Real-Valued Functions}} }} {{end-eqn}} {{qed|lemma}} === Distributivity over Vector Addition === {{begin-eqn}} {{eqn | l = \lambda \times_\R \map {\paren {f + g} } x       | r = \lambda \times_\R \paren {\map f x +_\R \map g x}       | c = {{Defof|Pointwise Addition of Real-Valued Functions}} }} {{eqn | r = \lambda \times_R \map f x +_\R \lambda \times_\R \map g x       | c = Real Multiplication Distributes over Addition }} {{eqn | r = \map {\paren{\lambda \cdot f} } x +_\R \map {\paren{\lambda \cdot g} } x       | c = {{Defof|Pointwise Scalar Multiplication of Real-Valued Functions}} }} {{eqn | r = \map {\paren {\lambda \cdot f + \mu \cdot f} } x       | c = {{Defof|Pointwise Addition of Real-Valued Functions}} }} {{end-eqn}} {{qed|lemma}} === Associativity with Scalar Multiplication === {{begin-eqn}} {{eqn | l = \map {\paren {\paren {\lambda \times_\R \mu} \cdot f} } x       | r = \paren {\lambda \times_\R \mu} \times_\R \map f x       | c = {{Defof|Pointwise Scalar Multiplication of Real-Valued Functions}} }} {{eqn | r = \lambda \times_\R \paren {\mu \times_\R \map f x}       | c = Real Multiplication is Associative }} {{eqn | r = \lambda \times_\R \map {\paren {\mu \cdot f} } x       | c = {{Defof|Pointwise Scalar Multiplication of Real-Valued Functions}} }} {{eqn | r = \map {\paren {\lambda \cdot \paren {\mu \cdot f} } } x       | c = {{Defof|Pointwise Scalar Multiplication of Real-Valued Functions}} }} {{end-eqn}} {{qed|lemma}} === Identity for Scalar Multiplication === {{begin-eqn}} {{eqn | l = \map {\paren {1 \cdot f} } x        | r = 1 \times_\R \map f x        | c = {{Defof|Pointwise Scalar Multiplication of Real-Valued Functions}} }} {{eqn | r = \map f x }} {{end-eqn}} {{qed}}