$C(\textbf{R})$ is a vector subspace of $\textbf{R}^\textbf{R}$? How do I see that $C(\textbf{R})$ is a vector subspace of $\textbf{R}^\textbf{R}$?

Let $\mathcal V$ be a vector space over a field $\mathbb F$. To check if a subset $\mathcal W\subset \mathcal V$ is a subspace of $\mathcal V$ it is sufficient to check:


*

*$0\in \mathcal W$

*$v,w\in\mathcal W\implies v+w\in\mathcal W$ for all $v,w\in\mathcal W$

*$v\in\mathcal W, a\in\mathbb F\implies a\cdot v\in\mathcal W$ for all $v\in\mathcal W,a\in\mathbb F$


In this case you have $\mathcal V=\mathbf{R}^{\mathbf{R}}$ and $\mathcal W=C(\mathbf{R})$ (which I assume to be the set of all continuous functions $f:\mathbf{R}\rightarrow\mathbf{R}$). Now apply your knowledge about continuous functions (e.g. the sum of two continuous functions) to check if $\mathcal W$ is a subspace of $\mathcal V$.
