Subspace of Smooth Real Functions
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Corollary to Subspace of Real Functions of Differentiability Class
Let $\mathbb J = \set {x \in \R: a < x < b}$ be an open interval of the real number line $\R$.
Let $\map {C^\infty} {\mathbb J}$ denote the set of all continuous real functions on $\mathbb J$ which are differentiable on $\mathbb J$ at all orders.
Then $\struct {\map {C^\infty} {\mathbb J}, +, \times}_\R$ is a subspace of the $\R$-vector space $\struct {\R^{\mathbb J}, +, \times}_\R$.
Proof
Note that by the definition of smooth real function:
- $\ds \map {C^\infty} {\mathbb J} = \bigcap_{m \mathop = 0}^\infty \map {C^m} {\mathbb J}$
By Subspace of Real Functions of Differentiability Class:
- $\struct {\map {C^m} {\mathbb J}, +, \times}_\R$ is a subspace of $\struct {\R^{\mathbb J}, +, \times}_\R$ for all $m$.
Then, from Set of Linear Subspaces is Closed under Intersection:
- $\ds \struct {\map {C^\infty} {\mathbb J}, +, \times}_\R$ 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 $27.5$