Question

In each of the following cases, either prove that the given subset W is a subspace of
V , or show why it is not a subspace of V .
a) \[V= c ^\infty (\mathbb{R})\] , W is the set of functions f⊆V for which
lim f(x) = 0
\[x \rightarrow \infty\].

Answer 1

Can you explain what \(e^{\infty} (\mathbb{R})\) is?

Answer 2

its not e, it's C . like, constant i believe?

Answer 3

its a vector space.

Answer 4

I'm still confused as to what \(c^{\infty}\) means. I've never seen notation like that.

Answer 5

i think it's like all of constants .. like constant to infinity vector space or something .. for all real numbers? heres to guessing hahaha.

Answer 6

In any case however, I think W is still a subspace because it contains the function \(f(x)=0\), W is closed under addition, and it's closed under scalar multiplication.

Answer 7

See http://en.wikipedia.org/wiki/Linear_subspace for more information.

Answer 8

ok thanks! one more question if you dont mind
same question applied .. but to
V=c∞(R)
, W is the set of functions f⊆V for which f(4) = 1

Answer 9

Definitely not. It doesn't contain the function \(f(x)=0\), it isn't closed under addition, and it isn't closed under scalar multiplication.

Answer 10

ok, are you able to explain to me sort of what i'm looking for here, because there are 2 other parts to this same question but i'm unsure how to define them

Answer 11

First, see if the function \(f(x)=0\) is in the subset.

Then, take two functions \(f, g\) in the subset. Find if the function \(h=f+g\) is in the subset as well.

Finally, take a function \(f\) in the subset, and some constant \(c\). Then find if \(c\cdot f\) is in the subspace.

If all three of these conditions are true, you have a subspace.

Answer 12

ok thank you so much!!!!!

Answer 13

You're welcome.