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
is a subspace of if and only if is closed under addition and closed under scalar multiplication.
Now I've two questions.
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 ?
I've the same question regarding and . Can we say is a subspace of ? If no then why not?