What is the dual $X^{*}$ of a finite-dimensional real vector space $X$ ? If $X$ has a basis $e_{1}, \ldots, e_{n}$, define the dual basis, and prove that it is indeed a basis of $X^{*}$.

[No results on the dimension of duals may be assumed without proof.]

Write down (without making a choice of basis) an isomorphism from $X$ to $X^{* *}$. Prove that your map is indeed an isomorphism.

Does every basis of $X^{*}$ arise as the dual basis of some basis of $X ?$ Justify your answer.

A subspace $W$ of $X^{*}$ is called separating if for every non-zero $x \in X$ there is a $T \in W$ with $T(x) \neq 0$. Show that the only separating subspace of $X^{*}$ is $X^{*}$ itself.

Now let $X$ be the (infinite-dimensional) space of all real polynomials. Explain briefly how we may identify $X^{*}$ with the space of all real sequences. Give an example of a proper subspace of $X^{*}$ that is separating.