There must be a mistake in my reasoning, but I couldn't find it and that's what I ask you to do:
Let $E_0 \subset E$ be a subspace in a normed space $E$ and $f: E_0 \rightarrow \mathbb{R}$ is continous functional on $E_0$
Let $\{e_0^1, e_0^2, …\}$ be a basis in $E_0$. We can extend it to the basis of entire $E$ using the Zorn's lemma: let $\{e_1^1, e_1^2,…\}$ be a system that $span \{e_0^1, e_0^2, …, e_1^1, e_2^2, …\}=E$
Let's define $f(e_1^i)=0$ for all $i$
So we've extended continous functional from subspace $E_0$ to the entire space $E$. But it's pretty much easier than how we do it in Hahn–Banach theorem so it must be a mistake somewhere
Maybe the extenstion will not be continious or it's not correct. But I can't understand. Need help to find a mistake
Best Answer
Such an extension isn't necessarily continuous.
For an example, consider the canonical (topological) basis $(e_n)_n$ of $\ell^2$ and let $B$ be a Hamel basis for $\ell^2$ which contains all $e_n$.
Pick $b_0 \in B$ different from all $e_n$ and define a continuous functional $f : \operatorname{span}\{b_0\} \to \mathbb{R}$ as $f(\alpha b_0) = \alpha$.
Extend $f$ to a functional $\ell^2 \to \mathbb{R}$ by setting $f(b) = 0, \forall b \in B\setminus \{b_0\}$. In particular $f(e_n) = 0, \forall n\in\mathbb{N}$.
If $f$ were continuous, we would have $f \equiv 0$, but clearly $f(b_0) = 1$. Hence $f$ is not continuous.
To be more explicit, you could take $b_0 = \sum_{n=1}^\infty \frac1n e_n$ as it is linearly independent with all $e_n$, and then let $B$ be a Hamel basis containing $\{b_0\} \cup \{e_n\}_{n\in\mathbb{N}}$.