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?