Subsets & Subspace Homework: Proofs & Counterexamples

[ak123456]

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 R^R, f is differentiable and f'(0) = 0

V:= fR^R, f is polynomial of the form f=at^2 for some aR
= 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

U: i am not sure about 0 , is it differentiable if f=0
V,W: is that 0 belongs to the polynomial, in other words i am not sure about the definition of polynomial
X: 0 is an odd function ? I knew that odd+odd=odd function a*odd=odd function

 

[Mark44]

U -- f(x) = 0 is differentiable, and its derivative is 0.
V, W -- a zero polynomial, such as 0 + 0t + 0t^2, is perfectly valid.
X - The 0 function is both even and odd (since f(-x) = f(x), it's even, and since f(-x) = -f(x), it's odd).

By the way, your vector space is R X R, I believe, not R^R

 

[ak123456]

U -- f(x) = 0 is differentiable, and its derivative is 0.
V, W -- a zero polynomial, such as 0 + 0t + 0t^2, is perfectly valid.
X - The 0 function is both even and odd (since f(-x) = f(x), it's even, and since f(-x) = -f(x), it's odd).

By the way, your vector space is R X R, I believe, not R^R

oh i c, thanks a lot