# 4.II.15F

Define the dual space $V^{*}$ of a finite-dimensional real vector space $V$, and explain what is meant by the basis of $V^{*}$ dual to a given basis of $V$. Explain also what is meant by the statement that the second dual $V^{* *}$ is naturally isomorphic to $V$.

Let $V_{n}$ denote the space of real polynomials of degree at most $n$. Show that, for any real number $x$, the function $e_{x}$ mapping $p$ to $p(x)$ is an element of $V_{n}^{*}$. Show also that, if $x_{1}, x_{2}, \ldots, x_{n+1}$ are distinct real numbers, then $\left\{e_{x_{1}}, e_{x_{2}}, \ldots, e_{x_{n+1}}\right\}$ is a basis of $V_{n}^{*}$, and find the basis of $V_{n}$ dual to it.

Deduce that, for any $(n+1)$ distinct points $x_{1}, \ldots, x_{n+1}$ of the interval $[-1,1]$, there exist scalars $\lambda_{1}, \ldots, \lambda_{n+1}$ such that

$\int_{-1}^{1} p(t) d t=\sum_{i=1}^{n+1} \lambda_{i} p\left(x_{i}\right)$

for all $p \in V_{n}$. For $n=4$ and $\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right)=\left(-1,-\frac{1}{2}, 0, \frac{1}{2}, 1\right)$, find the corresponding scalars $\lambda_{i}$.