## Month: November, 2012

### Advanced Analysis, Notes 14: Banach spaces (application: the Stone–Weierstrass Theorem revisited; structure of C(K))

In this post we will use the Krein–Milman theorem together with the Hahn–Banach theorem to give another proof of the Stone–Weierstrass theorem. The proof we present does not make use of the classical Weierstrass approximation theorem, so we will have here an alternative proof of the classical theorem as well.

### Advanced Analysis, Notes 13: Banach spaces (convex hulls and the Krein–Milman theorem)

It would be strange to disappear for a week without explanations. This blog was not working for the past week because of the situation in Israel. The dedication in the beginning of the previous post had something to do with this, too. We are now back to work, with the modest hope that things will remain quiet until the end of the semester. We begin our last chapter in basic functional analysis, convexity and the Krein–Milman theorem.

### Advanced Analysis, Notes 12: Banach spaces (application: existence of Haar measure)

This post is dedicated to my number one follower for her fourteenth birthday,which I spoiled…

Let $G$ be a compact abelian group. By this we mean that $G$ is at once both an abelian group and a compact Hausdorff topological space, and that the group operations are continuous, meaning that $g \mapsto g^{-1}$ is continuous on $G$ and $(g,h) \mapsto g+h$ is continuous as a map from $G \times G$ to $G$. It is known that there exists a regular Borel measure $\mu$ on $G$, called the Haar measure, which is non-negative, satisfies $\mu(G) = 1$, and is translation invariant:

$\forall g \in G . \mu(g+ E) = \mu(E) ,$

for every Borel set $E \subseteq G$. In fact, the Haar measure is known to exist in greater generality ($G$ does not have to be commutative and if one allows $\mu$ to be infinite then $G$ can also be merely locally compact). The Haar measure is an indispensable tool in representation theory and in ergodic theory. In this post we will use the weak* compactness of the unit ball of the dual to give a slick proof of the existence of the Haar measure in the abelian compact case.

Let $X$ be the Banach space $C([0,1])$ of continuous functions on the interval $[0,1]$ with the sup norm. Consider the following sequence of functions $\{f_n\}$ defind as follows. $f(0) = 0$ and $f_n(1/(n+1)) = 1$ for all $n = 1, 2, \ldots$,  $f_n$ is equal to zero on the interval between $2/(n+1)$ and $1$, and $f_n$ is linear in the intervals where we haven’t defined it yet (visualize!). The sequence is tending to zero pointwise, but the norm of $X$ does not detect this. The sequence tends to $0$ in the $L^1$ norm, but the $L^1$ norm is not in the game. Can the Banach space structure of $X$ detect the convergence of $f_n$ to $0$? Read the rest of this entry »