Exercise 11

Decide if each is a subspace of the vector space of real-valued functions of one real variable.

• The even functions $\{\hskip 1.00006pt\text{$f\colon\mathbb{R}\to\mathbb{R}$}\mid f(-x)=f(x)\text{for all }x\hskip 1.00006pt\}$. For example, two members of this set are $f_{1}(x)=x^{2}$ and $f_{2}(x)=\cos(x)$.

• The odd functions $\{\hskip 1.00006pt\text{$f\colon\mathbb{R}\to\mathbb{R}$}\mid f(-x)=-f(x)\text{ for all }x\hskip 1.00006pt\}$. Two members are $f_{3}(x)=x^{3}$ and $f_{4}(x)=\sin(x)$.