Some questions on 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 $x\in M$.

I am trying to write the proof as my homework but I meet some questions. Basically, I followed the proof on here: https://www.impan.pl/~lpalmisano/fa2017/lecture6.pdf. But I have two questions about the details in the proof.

1. In the first part of the 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 $f_1$ is a linear extension of $f$ with $f_1(x)\leqslant q(x)$ for all $x\in M_1$. Then how to get $S\neq\emptyset$? Is it because $(M,f)\in S$?

2. In the second part of the proof, we fix $\alpha\in\mathbb{R}$ and define define $F:X\rightarrow\mathbb{R}$ by $F(tx_0+y)=t\alpha+f(y)$ ($t\in\mathbb{R},y\in M$). How to show $F$ is well defined? And my attempt is: If $y_1=y_2\in M$ and $t_1=t_2\in\mathbb{R}$, then $$F(t_1x_0+y_1)=t_1\alpha+f(y_1)=t_2\alpha+f(y_2)=F(t_2x_0+y_2).$$ Thus $F:X\rightarrow\mathbb{R}$ is well defined. But my professor mentioned this part was wrong.

Any help will be appreciated!!

Answer 1

1. Yes.

2. Note that you are not defining $F$ on $X$, but on $\mathbb R x_0+M$. If you assume that $y_1=y_2$ and $t_1=t_2$, then there is nothing to prove. What you have to do is to assume that $t_1x_0+y_1=t_2x_0+y_2$. This you can rewrite as $$ (t_1-t_2)x_0=y_2-y_1. $$ Since $x_0\not\in M$ (that's the point, to extend from $M$, so $x_0\not\in M$) you get $t_1-t_2=0$. And then you get $y_2-y_1=0$. That is, there is a single $t$ and a single $y$ to express $tx_0+y\in \mathbb R x_0+M$. Thus $F$ is well-defined.