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
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.1$: Vector Space