Subspace of Real Continuous 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 $\map C {\mathbb J}$ be the set of all continuous real functions on $\mathbb J$.
Then $\struct {\map C {\mathbb J}, +, \times}_\R$ is a subspace of the $\R$-vector space $\struct {\R^{\mathbb J}, +, \times}_\R$.
Proof
By definition, $\map C {\mathbb J} \subseteq \R^{\mathbb J}$.
Let $f, g \in \map C {\mathbb J}$.
By Two-Step Vector Subspace Test, it needs to be shown that:
- $(1): \quad f + g \in \map C {\mathbb J}$
- $(2): \quad \lambda f \in \map C {\mathbb J}$ for any $\lambda \in \R$
$(1)$ follows by Sum Rule for Continuous Real Functions.
$(2)$ follows by Multiple Rule for Continuous Real Functions.
Hence $\struct {\map C {\mathbb J}, +, \times}_\R$ is a subspace of the $\R$-vector space $\struct {\R^{\mathbb J}, +, \times}_\R$.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases: Example $27.5$