Subspace of Riemann Integrable Functions
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Theorem
Let $\mathbb J = \set {x \in \R: a \le x \le b}$ be a closed interval of the real number line $\R$.
Let $\map \RR {\mathbb J}$ be the set of all Riemann integrable functions on $\mathbb J$.
Then $\struct {\map \RR {\mathbb J}, +, \times}_\R$ is a subspace of the $\R$-vector space $\struct {\R^{\mathbb J}, +, \times}_\R$.
Proof
Note that by definition, $\map \RR {\mathbb J} \subseteq \R^{\mathbb J}$.
Let $f, g \in \map \RR {\mathbb J}$.
Let $\lambda \in \R$.
By Linear Combination of Definite Integrals:
- $f + \lambda g$ is Riemann integrable on $\mathbb J$.
That is:
- $f + \lambda g \in \map \RR {\mathbb J}$
So by One-Step Vector Subspace Test:
- $\struct {\map \RR {\mathbb J}, +, \times}_\R$ is a subspace of $\struct {\R^{\mathbb J}, +, \times}_\R$.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases: Example $27.5$