Test Function Space with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space

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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


Sources

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