My book says:

The continuous functions

on the interval

form a subspace of

since they are closed under addition and scalar multiplication. To take this step further, for each positive

integer

, the functions with continuous

th derivatives on

form a subspace of

. So

is a subspace of

By definition:

is a subspace of

if and only if

is closed under addition and closed under scalar multiplication.

Now I've two questions.

1st one:

Why

is a subspace of

and not the other way around?

I mean I understand because they are closed under addition and scalar multiplication. But so is

. So can we say

is a subspace of

?

2nd one:

I've the same question regarding

and

. Can we say

is a subspace of

? If no then why not?