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

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Theorem

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

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


Proof

Let $f, g, h \in \map C I$ 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, $f + g \in \map C I$

$\Box$


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
\(\ds \) \(=\) \(\ds \map f x\)

$\blacksquare$


Sources

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