## Hahn-Banach Theorem: Crown Jewel of Functional Analysis

Hahn-Banach Theorem is called the Crown Jewel of Functional Analysis, and has many different versions.

There is a Chinese quote “实变函数学十遍，泛函分析心犯寒”, which means one needs to study real function theory ten times before understanding, and the heart can go cold when studying functional analysis, which shows how deep is this subject.

The following is one version of Hahn-Banach Theorem that I find quite useful:

(Hahn-Banach, Version) If $V$ is a normal vector space with linear subspace $U$ (not necessarily closed) and if $z$ is an element of $V$ not in the closure of $U$, then there exists a continuous linear map $\psi:V\to K$ with $\psi(x)=0$ for all $x\in U$, $\psi(z)=1$, and $\|\psi\|=\text{dist}(z,U)^{-1}$.

Brief sketch of proof: Define $\phi:U+\text{span}\{z\}\to K$, $\phi(u+\lambda z)=\lambda$, and use the Hahn-Banach (Extension version).