Incorrect proof of Hahn Banach Theorem
Solution 1:
How do You even know that $\phi'$ is continuous? You need to ensure that there exists $C_1$ such that $||u||\leq C_1||w+u||$ for any $w\in W, u\in U$ and to get Hahn-Banach You need $C_1=1$. Your proof works in unitary spaces since You can always take for $W$ an orthogonal complement of $U$.