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**x3bnm** My book says:

The continuous functions $\displaystyle C(-\infty, \infty)$ on the interval $\displaystyle (-\infty, \infty)$ form a subspace of $\displaystyle F(-\infty, \infty)$

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

integer $\displaystyle m$, the functions with continuous $\displaystyle m$th derivatives on $\displaystyle (-\infty, \infty)$ form a subspace of $\displaystyle C^1(-\infty, \infty)$. So $\displaystyle C^m(-\infty, \infty)$ is a subspace of $\displaystyle C^1(-\infty, \infty)$

By definition:

$\displaystyle W$ is a subspace of $\displaystyle V$ if and only if $\displaystyle W$ is closed under addition and closed under scalar multiplication.

Now I've two questions.

1st one:

Why $\displaystyle C(-\infty, \infty)$ is a subspace of $\displaystyle F(-\infty, \infty)$ and not the other way around?

I mean I understand because they are closed under addition and scalar multiplication. But so is $\displaystyle F(-\infty, \infty)$. So can we say $\displaystyle F(-\infty, \infty)$ is a subspace of $\displaystyle C(-\infty, \infty)$?

2nd one:

I've the same question regarding $\displaystyle C^1(-\infty, \infty)$ and $\displaystyle C^m(-\infty, \infty)$. Can we say $\displaystyle C^1(-\infty, \infty)$ is a subspace of $\displaystyle C^m(-\infty, \infty)$? If no then why not?