The Hahn-Banach Theorem

Complete the proof of Lemma 6.3.

Give the details of the proof of Corollary 6.5.

Show that $c^*$ is isometrically isomorphic to $l^1$. Are $c$ and $c_0$ isometrically isomorphic?

If $\mu$ is a Borel measure on $[0,1]$ and $\int x^n \, d\mu (x) = 0$ for all $n \geq 0$, show that $\mu = 0$.

If $n \geq 1$, show that there is a measure $\mu$ on $[0,1]$ such that for every polynomial $p$ of degree at most $n$,

If $n \geq 1$, does there exist a measure $\mu$ on $[0,1]$ such that $p'(0) = \int p \, d\mu$ for every polynomial of degree at most $n$?

Does there exist a measure $\mu$ on $[0,1]$ such that $\int p \, d\mu = p'(0)$ for every polynomial $p$?

Let $K$ be a compact subset of $\mathbb {C}$ and define $A(K)$ to be $\left\{ f \in C(K) : f \text{ is analytic on int } K\right\}$. (Functions here are complex valued.) Show that if $a \in K$, then there is a probability measure $\mu$ supported on $\partial K$ such that $f(a) = \int f \, d\mu$ for every $f$ in $A(K)$. (A probability measure is a nonnegative measure $\mu$ such that $\left\| \mu \right\| = 1$.)

If $K = \mathrm{cl}\, \mathbb {D}$ ($\mathbb {D} = \left\{ \left|z\right| < 1\right\}$) and $a \in K$, find the measure $\mu$ whose existence was proved in Exercise 8.

Let $P = \left\{ p|\partial \mathbb {D} : p = \text{an analytic polynomial}\right\}$ and consider $P$ as a manifold in $C(\partial \mathbb {D})$. Show that if $\mu$ is a real-valued measure on $\partial \mathbb {D}$ such that $\int p \, d\mu = 0$ for every $p$ in $P$, then $\mu = 0$. Give an example of a complex-valued measure $\mu$ such that $\mu \neq 0$ but $\int p \, d\mu = 0$ for every $p$ in $P$.