Are These Subsets of R^R Subspaces?

[karnten07]

Homework Statement

Which of the following subsets of the vector space R^R of all functions from R to R are subspaces? (proofs or counterexamples required)

U:= f \inR^R, f is differentiable and f'(0) = 0

V:= f\inR^R, f is polynomial of the form f=at^2 for some a\inR
= There exists a of the set R: for all s of R: f(s) = as^2

W:= " " f is polynomial of the form f=at^i for some aof the set R and i of the set N
= there exists i of N, there exists a of R: that for all s of R: f(s) = as^i

X:= " " f is odd
(f is odd such that f(-s) =-f(s) for all s of R

Homework Equations

The Attempt at a Solution

Okay so i want to start with , odd functions. I can use the sine function as a counterexample because i don't think X is a subspace. I think that it isn't closed under addtion because sin90 +sin90 = 2 which isn't a solution to any elements of the set X. So i can use this as a counterexample right?

I will start thinking about the other subsets.

 

[Dick]

Gack! The question doesn't ask you whether a single odd function forms a subspace, it asks you whether ALL odd functions form a subspace. I.e. you have to prove things like if f(s) and g(s) are odd, that f(s)+g(s) is odd and k*f(s) is odd, right?

 

[karnten07]

Gack! The question doesn't ask you whether a single odd function forms a subspace, it asks you whether ALL odd functions form a subspace. I.e. you have to prove things like if f(s) and g(s) are odd, that f(s)+g(s) is odd and k*f(s) is odd, right?

yes, if i want to show the subset is a subspace. But if i want to show that it is not a subset as one function in the subset doesn't meet the criteria i can use a counterexample, right? So for this one, i can make it easy in doing it that way.

 

[benorin]

You are on the right track checking closure properties: you must have closure under scalar multiplication and closure under vector addition for a subset to be subspace. Your example of sin90 + sin90 = 2 didn't start with two vectors in X (sin90 = 1 is not an odd function).

Note: Here the vectors are functions, so testing closure under vector addition for this problem looks like this:

Let f(x) and g(x) be vectors (functions) in the subset X.
Then f(x)+g(x) is (or is not) a vector in the subset X because...
<Put some stuff here>
Hence U is (or is not) closed under vector addition.

 

[Dick]

No! For one thing the odd functions ARE a subspace. Look, the real numbers are a vector space, right? Just because 1+1 is not equal to 1 doesn't prove they aren't. A subspace is a group of objects and closure just says that combinations of them remain in the same group. You can't pick one out and form a contradiction solely on that.

 

[Dick]

Maybe this is what is confusing you. sin(x)+sin(x)=2*sin(x) NOT sin(x+x). All of those are odd, and sin(90)+sin(90)=2*sin(90). There is absolutely nothing in the definition of subspace that says you have to be able to solve sin(90)+sin(90)=sin(y). Is there?

 

[karnten07]

Maybe this is what is confusing you. sin(x)+sin(x)=2*sin(x) NOT sin(x+x). All of those are odd, and sin(90)+sin(90)=2*sin(90). There is absolutely nothing in the definition of subspace that says you have to be able to solve sin(90)+sin(90)=sin(y). Is there?

Oh yes, i see what you are saying now. These questions takes ages to write out ugh.

 

[karnten07]

You are on the right track checking closure properties: you must have closure under scalar multiplication and closure under vector addition for a subset to be subspace. Your example of sin90 + sin90 = 2 didn't start with two vectors in X (sin90 = 1 is not an odd function).

Note: Here the vectors are functions, so testing closure under vector addition for this problem looks like this:

Let f(x) and g(x) be vectors (functions) in the subset X.
Then f(x)+g(x) is (or is not) a vector in the subset X because...
<Put some stuff here>
Hence U is (or is not) closed under vector addition.

mmm I am confused. How do i show f(x) + g(x) is still odd? I don't get the vector addition part either??

 

[HallsofIvy]

The point Dick was making is that you are wrong. The sum of two odd functions is an odd function and any number times an odd function is an odd function. The set of all odd functions is a subspace of R^R. What you need to do is to prove the two statements in the second sentence.