Let $C( \mathbb{R}) $ be the collection of all continuous functions from $\mathbb{R} $ to $\mathbb{R} $ . Then $C( \mathbb{R}) $ is a real vector space with pointwise addition and scalar multiplication defined by

(f+g)(x) = f(x)+g(x) and (rf)(x) = rf(x)

for all f, g $ \in C( \mathbb{R}) $ and all r, x $ \in \mathbb{R} $. Which of the following are subspaces of $ C( \mathbb{R}) $?

Question

Let $C( \mathbb{R}) $ be the collection of all continuous functions from  $\mathbb{R} $ to  $\mathbb{R} $ . Then $C( \mathbb{R}) $ is a real vector space with pointwise addition and scalar multiplication defined by 

(f+g)(x) = f(x)+g(x) and (rf)(x) = rf(x)

for all f, g $ \in C( \mathbb{R}) $ and all r, x $ \in  \mathbb{R} $. Which of the following are subspaces of $  C( \mathbb{R}) $?

kainui · Answer

I. { f : f is twice differentiable and f''(x)-2f'(x)+3f(x)=0 for all x} 

II. { g : g is twice differentiable and g''(x)=3g'(x) for all x}

III. { h : h is twice differentiable and h''(x)=h(x)+1 for all x}

So there may be multiple correct answers here. I don't understand what's going on here exactly.