Apologies for any bad formatting.
This is number 35 from chapter 3.F of Axler's Linear Algebra Done Right, if you have the text to view.
I've been stuck on this one for quite some time: obviously because the vector spaces are infinite dimensional, we cannot create the isomorphism using bases of the spaces. Also, because $\Bbb{R}^\infty$ contains infinite sequences, it is not clear to me how to create an injective function...
I also thought that maybe I could develop the isomorphism using finite dimensional subspaces of the respective spaces, which is relatively easy, but being able to abstract that iso for the infinite dimensional spaces is beyond me.
Any assistance/explanation would be greatly appreciated.
As with all algebraic duals, the correspondence is based on looking at how each functional behaves on a basis for the original vector space (in your case, polynomials). So let $\beta = \{1, t, t^2, \ldots\}$ be the standard monomial basis for the space of real polynomials. Then, by linearity, every linear functional $L \in (P(R))'$ is uniquely defined by its values on $\beta$, namely $L(1), L(t), L(t^2), \ldots$. Perhaps, more suggestively, let $L(1) = a_0, L(t) = a_1, \ldots, L(t^k) = a_k$. Then $L$ is determined by a set $(a_0, a_1, a_2, \ldots)$. Because it sounds like you are a new student, it might be helpful that I leave some remaining details for you to work out, namely making sure that this correspondence is both injective and surjective.