Perhaps I should be more specific in my confusion.

In a problem that I am working on I am asked a question involving the vector space $\displaystyle V=P( \mathbb{R} )$, the vector space of polynomials with coefficients from $\displaystyle \mathbb{R}$.

I know that $\displaystyle \beta = \{ 1, x, x^2, ... \} $ is a basis for $\displaystyle V$.

If I blindly apply the definition of a Dual Basis (which I mentioned previously) then I know,

$\displaystyle f_1(1)=1$

$\displaystyle f_1(x)=0$

...

and $\displaystyle f_2(1)=0$

$\displaystyle f_2(x)=1$

$\displaystyle f_2 \left( x^2 \right) = 0$

and so on...

What does this mean?

I don't even know if knowing what $\displaystyle V^*$ is will help me with this problem but I want to know.

Does this mean that $\displaystyle f_1$ is some linear functional that takes the first vector in my basis and maps it to 1 and every other vector in the basis to zero?

What does $\displaystyle V^*$ look like??

Please help.

Thank you in advance too!