# Thread: Subspace of a Vector space: Why not the other way around?

1. ## Subspace of a Vector space: Why not the other way around?

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?

2. Originally Posted by 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?
Is the space $\displaystyle F(-\infty, \infty)$ the set of real function?
If so isn't it true that $\displaystyle C(-\infty, \infty)\subset F(-\infty, \infty)$ and not the other way?

Maybe you should define the symbols.

3. Originally Posted by Plato
Is the space $\displaystyle F(-\infty, \infty)$ the set of real function?
If so isn't it true that $\displaystyle C(-\infty, \infty)\subset F(-\infty, \infty)$ and not the other way?

Maybe you should define the symbols.
Yes $\displaystyle F(-\infty, \infty)$ is a set of real function. I understand what you're saying.

$\displaystyle C^1(-\infty, \infty)$ = set of continuous first derivative
$\displaystyle C^m(-\infty, \infty)$ = set of continuous m derivatives

But can we say(described in above post) $\displaystyle C^1(-\infty, \infty)$ be a subspace of $\displaystyle C^m(-\infty, \infty)$?

4. I think you're missing a major part of the definition of a subspace (namely, the SUB part).

The definition is: $\displaystyle W$ is a subspace of $\displaystyle V$ means that both $\displaystyle V,W$ are vector spaces, AND THAT $\displaystyle W\subseteq V$.

5. Originally Posted by topspin1617
I think you're missing a major part of the definition of a subspace (namely, the SUB part).

The definition is: $\displaystyle W$ is a subspace of $\displaystyle V$ means that both $\displaystyle V,W$ are vector spaces, AND THAT $\displaystyle W\subseteq V$.
Thanks for explaining.