The proof presented in my Analysis Book seemed a little complicated, and I'm wondering if we can simplify it using tools of Algebra.
$\textbf{Proof:}$ Let $W\leq X$ where $X$ is a vector space over $\mathbb{R}.$ Let $f:W\to\mathbb{R}$ be a linear Functional, and suppose that $p:X\to \mathbb{R}$ is a linear functional such that $f\leq p|_W.$ Since $V$ is a module over a semisimple ring it is semisimple, hence there exists $M\leq V$ with $V=M\oplus W.$ Then we define $F:V\to \mathbb{R}$ via the universal Property of the Direct Sum to be the unique linear map such that $F|_W=f$ and $F|_M=p|_M.$ Here it is clear $F$ extends $f.$ Now if $x\in X$ then there exists unique $w\in W$ and $m\in M$ with $x=w+m,$ so $$F(x)=F(w+m)=F(w)+F(m)=f(w)+p(m)\leq p(w)+p(m)=p(w+m)=p(x).$$ Which completes the proof. $\blacksquare$
Does the proof check out? Any advice? Thank you.