Proof of $\text{Span}(A \cup \{x_0\})$ for $x_0 \notin A$ can uniquely determine $x = m + ax_0$? I have a difficult time understanding the claim $x = m + ax_0$ as stated in below text (it regards the proof of the Hahn-Banach theorem). I would rather look at a proof or a beginning of a proof; could anyone help me out a bit? It would be really appreciated.


The condition $x_0 \in E \setminus M$ basically means that $M$ and $\operatorname{span}\{x_0\}$ are linearly independent, and then the condition $E = \operatorname{span}(M \cup \{x_0\})$ means that $E$ is the internal direct sum of $M$ and $\operatorname{span}\{x_0\}$, so each element of $E$ can be represented uniquely as an element of $M$ plus an element of $\operatorname{span}\{x_0\}$, i.e. as $m + ax_0$ where $m \in M$ and $a \in \Bbb R$.
