I am reading "Linear Algebra Done Right 3rd Edition" by Sheldon Axler.
1.22 Example $\mathbb{F}^\infty$ is defined to be the set of all sequences of elements of $\mathbb{F}$: $$\mathbb{F}^\infty=\{(x_1,x_2,\dots) : x_j\in\mathbb{F}\text{ for }j=1,2,\dots\}.$$
2.11 Definition polynomial, $\mathcal{P}(\mathbb{F})$
- A function $p:\mathbb{F}\to\mathbb{F}$ is called a polynomial with coefficients in $\mathbb{F}$ if there exist $a_0,\dots,a_m\in\mathbb{F}$ such that $$p(z)=a_0+a_1z+a_2z^2+\dots+a_mz^m$$ for all $z\in\mathbb{F}.$
- $\mathcal{P}(\mathbb{F})$ is the set of all polynomials with coefficients in $\mathbb{F}.$
3.94 Definition dual space, $V'$
The dual space of $V$, denoted $V'$, is the vector space of all linear functionals on $V$. In other words, $V'=\mathcal{L}(V,\mathbb{F})$.
Exercise 3.F.35
Show that $(\mathcal{P}(\mathbb{R}))'$ and $\mathbb{R}^\infty$ are isomorphic.
I tried to solve this exercise, but I was not able to solve this exercise.
I know the following facts:
$\varphi:\mathcal{P}(\mathbb{R})\ni p\mapsto p^{(n)}(a)\in\mathbb{R}$ is a linear functional, where $a\in\mathbb{R}$ and $p^{(n)}$ is the $n$th derivative of $p$.
$\varphi:\mathcal{P}(\mathbb{R})\ni p\mapsto \int_{a}^{b} p(x) dx\in\mathbb{R}$ is a linear functional, where $a,b\in\mathbb{R}$.