Let $E$ be a Banach Space, $F \subset E$ non-closed subspace, $T \in L(F,F)$ the identity. Then there is no $S \in L(E,F)$ extending $T$. Why?
2026-04-18 23:14:57.1776554097
Why Hahn-Banach can not be generalized to operators
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The space $F$ is non-complete, so there are “not enough space” to place the continuous images of the points of its completion $\bar{F}\subset E$. The domain completeness (of some kind) is a typical condition in map extension theorems. The intuition here is similar to that we cannot extend the identity map $\operatorname{id}:\Bbb Q\to\Bbb Q$ from $\Bbb Q\subset \Bbb R$ to a continuous map from $\Bbb R$ to $\Bbb Q$.