Exercise 1.3.16

Question

Let $C^{n}(R)$ denote the set of all real-valued functions defined on the real line that have a continuous $n$th derivative. Prove that $\mathrm{C}^{n}(R)$ is a subspace of $\mathcal{F}(R, R)$.

Jephian C.-H. Lin · Accepted Answer

\begin{proof}
If $f^{(n)}$ and $g^{(n)}$ are the $n$th derivative of $f$ and $g$. Then $f^{(n)}+g^{(n)}$ will be the $n$th derivative of $f+g$. And it will continuous if both $f^{(n)}$ and $g^{(n)}$ are continuous. Similarly $cf^{(n)}$ is the $n$th derivative of $cf$ and it will be continuous. This space has zero function as the zero vector.
\end{proof}

Exercise 1.3.16

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Exercise 1.3.16

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