Test Function Space with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space
Theorem
Let $\map \DD {\R^d}$ be the test function space.
Let $\struct {\C, +_\C, \times_\C}$ be the field of complex numbers.
Let $\paren +$ be the pointwise addition of test functions.
Let $\paren {\, \cdot \,}$ be the pointwise scalar multiplication of test functions over $\C$.
Then $\struct {\map \DD {\R^d}, +, \, \cdot \,}_\C$ is a vector space.
Proof
Let $f, g, h \in \map \DD {\R^d}$ be test functions with the compact support $K$.
Let $\lambda, \mu \in \C$.
Let $\map 0 x$ be a real-valued function such that:
- $\map 0 x : \R^d \to 0$.
Let us use real number addition and multiplication.
$\forall x \in \R^d$ define pointwise addition as:
- $\map {\paren {f + g}} x := \map f x +_\C \map g x$.
Define pointwise scalar multiplication as:
- $\map {\paren {\lambda \cdot f}} x := \lambda \times_\C \map f x$
Let $\map {\paren {-f} } x := -\map f x$.
Closure Axiom
By Sum Rule for Continuous Complex Functions, $f + g \in \map \DD {\R^d}$
$\Box$
Commutativity Axiom
By Pointwise Addition on Complex-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
\(\ds \map {\paren {0 + f} } x\) | \(=\) | \(\ds \map 0 x +_\C \map f x\) | Definition of Pointwise Addition of Complex-Valued Functions | |||||||||||
\(\ds \) | \(=\) | \(\ds 0 +_\C \map f x\) | Definition of $\map 0 x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map f x\) |
$\Box$
Inverse Axiom
\(\ds \map {\paren {f + \paren {-f} } } x\) | \(=\) | \(\ds \map f x +_\C \map {\paren {-f} } x\) | Definition of Pointwise Addition of Complex-Valued Functions | |||||||||||
\(\ds \) | \(=\) | \(\ds \map f x +_\C \paren {-1} \times_\C \map f x\) | Definition of $\map {\paren {-f} } x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
$\Box$
Distributivity over Scalar Addition
\(\ds \map {\paren { \paren {\lambda +_\C \mu} f} } x\) | \(=\) | \(\ds \paren {\lambda +_\C \mu} \times_\C \map f x\) | Definition of Pointwise Scalar Multiplication of Complex-Valued Functions | |||||||||||
\(\ds \) | \(=\) | \(\ds \lambda \times_\C \map f x +_\C \mu \times_\C \map f x\) | Complex Multiplication Distributes over Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\paren {\lambda \cdot f} } x +_\C \map {\paren {\mu\cdot f} } x\) | Definition of Pointwise Scalar Multiplication of Complex-Valued Functions | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\paren {\lambda \cdot f + \mu \cdot f} } x\) | Definition of Pointwise Addition of Complex-Valued Functions |
$\Box$
Distributivity over Vector Addition
\(\ds \lambda \times_\C \map {\paren {f + g} } x\) | \(=\) | \(\ds \lambda \times_\C \paren {\map f x +_\C \map g x}\) | Definition of Pointwise Addition of Complex-Valued Functions | |||||||||||
\(\ds \) | \(=\) | \(\ds \lambda \times_R \map f x +_\C \lambda \times_\C \map g x\) | Complex Multiplication Distributes over Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\paren{\lambda \cdot f} } x +_\C \map {\paren{\lambda \cdot g} } x\) | Definition of Pointwise Scalar Multiplication of Complex-Valued Functions | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\paren {\lambda \cdot f + \mu \cdot f} } x\) | Definition of Pointwise Addition of Complex-Valued Functions |
$\Box$
Associativity with Scalar Multiplication
\(\ds \map {\paren {\paren {\lambda \times_\C \mu} \cdot f} } x\) | \(=\) | \(\ds \paren {\lambda \times_\C \mu} \times_\C \map f x\) | Definition of Pointwise Scalar Multiplication of Complex-Valued Functions | |||||||||||
\(\ds \) | \(=\) | \(\ds \lambda \times_\C \paren {\mu \times_\C \map f x}\) | Complex Multiplication is Associative | |||||||||||
\(\ds \) | \(=\) | \(\ds \lambda \times_\C \map {\paren {\mu \cdot f} } x\) | Definition of Pointwise Scalar Multiplication of Complex-Valued Functions | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\paren {\lambda \cdot \paren {\mu \cdot f} } } x\) | Definition of Pointwise Scalar Multiplication of Complex-Valued Functions |
$\Box$
Identity for Scalar Multiplication
\(\ds \map {\paren {1 \cdot f} } x\) | \(=\) | \(\ds 1 \times_\C \map f x\) | Definition of Pointwise Scalar Multiplication of Complex-Valued Functions | |||||||||||
\(\ds \) | \(=\) | \(\ds \map f x\) |
$\blacksquare$
Also see
- Complex Vector Space is Vector Space
- Space of Continuous on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 6.1$: A glimpse of distribution theory. Test functions, distributions, and examples