### Advanced Analysis, Notes 6: Banach spaces (basics, the Hahn-Banach Theorems)

Recall that a norm on a (real or complex) vector space $X$ is a function $\| \cdot \| : X \rightarrow [0, \infty)$ that satisfies for all $x,y \in X$ and all scalars $a$ the following:

1. $\|x\| = 0 \Leftrightarrow x = 0$.
2. $\|ax\| = |a| \|x\|$.
3. $\|x + y \| \leq \|x\| + \|y\|$.

A vector space with a norm on it is said to be a normed space. Inner product spaces are normed spaces. However, many norms of interest are not induced by an inner product. In fact:

Exercise A: A norm is induced by an inner product if and only if it satisfies the parallelogram law:

$\|x+y\|^2 + \|x-y\|^2 = 2 \|x\|^2 + 2\|y\|^2 .$

Instead of solving this exercise, you might prefer to read this old paper where Jordan and von Neumann prove this.

Using Exercise A, it is not hard to show that some very frequently occurring norms, such as the sup norm on $C(X)$ or the operator norm on $B(H)$, are not induced by inner products. The latter example shows that even if one is working in the setting of Hilbert spaces one is led to study other normed spaces. We now begin our study of normed spaces and, particular, Banach spaces.