Subspace of Riemann Integrable Functions

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 $27.5$