Reference request: Proof for Krein's extension theorem

Question

I want the proof of following theorems, which is the exact words from Zeidler's nonlinear functional analysis. Vol, III, page 171.

Suppose that the following two conditions hold. (i) $X$ is a real locally convex space. $K$ is a convex cone in $K$ and $L$ is a linear subspace of $X$ such that $L\cap int\,{K}\neq \emptyset$. (ii) $f:L\to \mathbb{R}$ is a linear functional such that $f(u)\geq 0$ for all $u\in L\cap K$. Then $f$ can be extended to a continuous linear functional $f:X\to \mathbb{R}$ such that $f(u)\geq 0$ all $u\in K$.

Does anyone know some exact references or notes that contain this theorem?

Answer 1

I have found a proof from "Real Analysis with Economic Applications" and would like to post here for anyone that may be interested in this problem. We introduce a preorder $\preceq $ in $X$ such that $y\preceq x $ if and only if $x\in y+ K$. Then we introduce a functional
$$ \varphi(x)=\inf\{f(y); x \preceq y,\,\, y\in L\}. $$ We need to show that it is well-defined for all $x$. Let $x_0\in int(K)\cap L$ and $U$ be any balanced neighbourhood of $0$ in $X$ such that $x_0+U\subset K$. Then for any $x$, there exits a $\lambda>0$ such that $\lambda x\in U$, then $-\frac{x}{\lambda}\in U$ and so $x_0-x/\lambda \in P$. Hence, $\varphi(x)\leq \lambda f(x_0)$. It is readily checked that $\varphi$ is subaddictive and positively homogeneous and $f=\varphi\mid_L$. By Hahn-Banach's theorem, we can extend $f$ into a linear functional $f^*$ such that $f^*(x)\geq 0$ for all $K$ and $|f(x)|\leq |\varphi(x)|\leq \lambda f(x_0)$, which implies the continuity of $f$.

Answer 2

See there, maybe it is useful: https://wikivisually.com/wiki/Krein_extension_theorem