Let $V_1=\{f: \mathbf{R}\rightarrow \mathbf{R}|f(x)=f(-x)$ for all $x\in\mathbf{R} \}$, how to check $\cos(x)$ and $x^2$ define functions on $V_1$?

Question

By theorem, $W$ is a subspace of $V$ iff for all $x, y \in W$, and all $c\in \mathbf{R}$, we have $cx+y\in W$.

But since $\cos(x)$ and $x^2$ are functions, how do you check by using the theorem?

Answer 1

The question is simply whether $\cos (x)$ and $x^2$ are elements of $V_1$. So you do not need to bother with subspaces at all. You just need to check whether $cos(x)=cos(-x)$ and $x^2=(-x)^2$ $\forall x \in \mathbb{R}$. As you should know both those functions are even so they do belong in $V_1$.