Originally Posted by

**Gojinn** I'm hoping this problem hasn't been posted before. I couldn't find a similar problem in the search so I'm posting a new thread.

Right, the following question was giving to us in one of our worksheets:

Now I know that, in order to prove that a certain vector space (V) is a subspace of another vector space (H), you have to go through the following three axioms:

1) The zero vector of V is in H.

Is it true that $\displaystyle \int\limits_{-1}^1\frac{0}{\sqrt{1-x^2}}dx=0$ ?

2) H is closed under vector addition.

Is it true that if $\displaystyle \int\limits_{-1}^1\frac{p(x)}{\sqrt{1-x^2}}dx=0\,\,\,and\,\,\,\int\limits_{-1}^1\frac{g(x)}{\sqrt{1-x^2}}dx=0$ then also $\displaystyle \int\limits_{-1}^1\frac{p(x)+g(x)}{\sqrt{1-x^2}}dx=0$ ?

3) H is closed under scalar multiplication.

Is is true that if $\displaystyle \int\limits_{-1}^1\frac{p(x)}{\sqrt{1-x^2}}dx=0$ then also $\displaystyle \int\limits_{-1}^1\frac{kp(x)}{\sqrt{1-x^2}}dx=0\,\,\,\forall\,k\in\mathbb{F}=$ the definition field?

Tonio

The bit that's confusing me is the whole "integration" thing. I'm assuming the the range between -1 and 1 is reducing the size of the space of V... so am I supposed to integrate first or... I dunno, I just need general help with this problem.