Verify the well-defined in the proof of Hahn Banach Theorem.

Question

Theorem: Let $q:X\rightarrow\mathbb{R}$ be a sublinear functional on a real linear space $X$. Let $M$ be a linear subspace of $X$ and suppose that $f:M\rightarrow\mathbb{R}$ is a linear functional such that $f(x)\leqslant q(x)$ for all $x\in M$. Then there exists a linear functional $F:X\rightarrow\mathbb{R}$ such that $F(x)\leqslant q(x)$ for all $x\in X$ and $F(x)=f(x)$ for all $x\in M$.

Proof: let $S$ be the collection of all pairs $(M_1,f_1)$, where $M_1$ is a linear subspace of $X$ such that $M\subseteq M_1$ and $f_1:M_1\rightarrow\mathbb{R}$ is a linear extension of $f$ satisfying $f_1(x)\leqslant q(x)$ for all $x\in M_1$. By the claim above, $S\neq\emptyset$. For any $(M_1,f_1),(M_2,f_2)\in S$, define $(M_1,f_1)\preccurlyeq(M_2,f_2)$ if $M_1\subseteq M_2$ and $f_2$ is a linear extension of $f_1$. Then $\preccurlyeq$ is a partial order on $S$. Let $C=\{(M_i,f_i):i\in I\}$ be a chain in $S$. Let $\tilde{M}=\bigcup\limits_{i\in I}M_i$. Then $\tilde{M}$ is a linear subspace of $X$ since each $M_i$ is a linear subspace of $X$. Define $\tilde{f}:\tilde{M}\rightarrow\mathbb{R}$ by $\tilde{f}(x)=f_i(x)$ if $x\in M_i$ for some $i\in I$.

My question is how to show $\tilde{f}$ is well defined? Should I pick $x,y\in\tilde{M}$ and show $\tilde{f}(x)=\tilde{f}(y)$?

Thank you!

Answer 1

Assume $x \in M_i \cap M_j$ for some $i, j \in I$. Since $C$ is a chain, we have $(M_i, f_i) \preccurlyeq (M_j, f_j)$ or $(M_j, f_j) \preccurlyeq (M_i, f_i)$.

Without loss of generality, assume $(M_i, f_i) \preccurlyeq (M_j, f_j)$.

Then $M_i \subseteq M_j$ and $f_j|_{M_i} = f_i$ so in particular $f_i(x) = f_j(x)$.

Hence, $\tilde f$ is well defined.

Answer 2

For $x,y\in\widetilde{M}$, $x=y$, say, $x\in M_{1}$ and $y\in M_{2}$, but $\{M_{i}\}$ is a chain, so, say, $M_{1}\subseteq M_{2}$, and hence $x,y\in M_{2}$ and $f_{2}(x)=f_{2}(y)$, but $f_{2}$ is an extension of $f_{1}$, so $f_{1}(x)=f_{2}(x)$, overall we have $f_{1}(x)=f_{2}(y)$, hence $\widetilde{f}(x)$ is defined in either $f_{1}(x)$ or $f_{2}(y)$ is no problem.