# 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, $f + g \in \map \CC I$

$\Box$

### Commutativity Axiom

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

$\Box$

### Associativity Axiom

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

$\Box$

### Identity Axiom

 $\ds \map {\paren {0 + f} } x$ $=$ $\ds \map 0 x +_\R \map f x$ Definition of Pointwise Addition of Real-Valued Functions $\ds$ $=$ $\ds 0 +_\R \map f x$ Definition of $\map 0 x$ $\ds$ $=$ $\ds \map f x$

$\Box$

### Inverse Axiom

 $\ds \map {\paren {f + \paren {-f} } } x$ $=$ $\ds \map f x +_\R \map {\paren {-f} } x$ Definition of Pointwise Addition of Real-Valued Functions $\ds$ $=$ $\ds \map f x +_\R \paren {-1} \times_\R \map f x$ Definition of $\map {\paren {-f} } x$ $\ds$ $=$ $\ds 0$

$\Box$

### Distributivity over Scalar Addition

 $\ds \map {\paren { \paren {\lambda +_\R \mu} f} } x$ $=$ $\ds \paren {\lambda +_\R \mu} \times_\R \map f x$ Definition of Pointwise Scalar Multiplication of Real-Valued Functions $\ds$ $=$ $\ds \lambda \times_\R \map f x +_\R \mu \times_\R \map f x$ Real Multiplication Distributes over Addition $\ds$ $=$ $\ds \map {\paren {\lambda \cdot f} } x +_\R \map {\paren {\mu\cdot f} } x$ Definition of Pointwise Scalar Multiplication of Real-Valued Functions $\ds$ $=$ $\ds \map {\paren {\lambda \cdot f + \mu \cdot f} } x$ Definition of Pointwise Addition of Real-Valued Functions

$\Box$

### Distributivity over Vector Addition

 $\ds \lambda \times_\R \map {\paren {f + g} } x$ $=$ $\ds \lambda \times_\R \paren {\map f x +_\R \map g x}$ Definition of Pointwise Addition of Real-Valued Functions $\ds$ $=$ $\ds \lambda \times_R \map f x +_\R \lambda \times_\R \map g x$ Real Multiplication Distributes over Addition $\ds$ $=$ $\ds \map {\paren{\lambda \cdot f} } x +_\R \map {\paren{\lambda \cdot g} } x$ Definition of Pointwise Scalar Multiplication of Real-Valued Functions $\ds$ $=$ $\ds \map {\paren {\lambda \cdot f + \mu \cdot f} } x$ Definition of Pointwise Addition of Real-Valued Functions

$\Box$

### Associativity with Scalar Multiplication

 $\ds \map {\paren {\paren {\lambda \times_\R \mu} \cdot f} } x$ $=$ $\ds \paren {\lambda \times_\R \mu} \times_\R \map f x$ Definition of Pointwise Scalar Multiplication of Real-Valued Functions $\ds$ $=$ $\ds \lambda \times_\R \paren {\mu \times_\R \map f x}$ Real Multiplication is Associative $\ds$ $=$ $\ds \lambda \times_\R \map {\paren {\mu \cdot f} } x$ Definition of Pointwise Scalar Multiplication of Real-Valued Functions $\ds$ $=$ $\ds \map {\paren {\lambda \cdot \paren {\mu \cdot f} } } x$ Definition of Pointwise Scalar Multiplication of Real-Valued Functions

$\Box$

### Identity for Scalar Multiplication

 $\ds \map {\paren {1 \cdot f} } x$ $=$ $\ds 1 \times_\R \map f x$ Definition of Pointwise Scalar Multiplication of Real-Valued Functions $\ds$ $=$ $\ds \map f x$

$\blacksquare$