Hahn-Banach Theorem/Real Vector Space/Lemma 2
Lemma
Let $X$ be a vector space over $\R$.
Let $p : X \to \R$ be a sublinear functional.
Let $X_0$ be a linear subspace of $X$.
Let $f_0 : X_0 \to \R$ be a linear functional such that:
- $\map {f_0} x \le \map p x$ for each $x \in X_0$.
Let $P$ be the set of pairs $\tuple {G, g}$ such that:
- $(1): \quad$ $G$ is a linear subspace of $X$ with $X_0 \subseteq G$
- $(2): \quad$ $g : G \to \R$ is a linear functional extending $f_0$
- $(3): \quad$ $\map g x \le \map p x$ for each $x \in G$.
Define the relation $\preceq$ on $P$ by:
- $\tuple {G, g} \preceq \tuple {H, h}$ if and only if:
- $(1): \quad$ $G \subseteq H$
- $(2): \quad$ $h$ extends $g$.
Then:
- $\struct {P, \preceq}$ is an ordered set.
Proof
We check each of the conditions of an ordering for $\preceq$.
Proof of $(1)$
We show that for each $\tuple {G, g} \in P$, we have:
- $\tuple {G, g} \preceq \tuple {G, g}$
We have, from Set is Subset of Itself:
- $G \subseteq G$
We also have:
- $g$ extends $g$.
So, we have:
- $\tuple {G, g} \preceq \tuple {G, g}$
$\Box$
Proof of $(2)$
Suppose that $\tuple {G, g}, \tuple {H, h}, \tuple {I, i} \in P$ have:
- $\tuple {G, g} \preceq \tuple {H, h}$
and:
- $\tuple {H, h} \preceq \tuple {I, i}$
We show that:
- $\tuple {G, g} \preceq \tuple {I, i}$
We have:
- $G \subseteq H$
and:
- $H \subseteq I$
From Subset Relation is Transitive, we have:
- $G \subseteq I$
Since $h$ extends $g$, we have:
- $\map g x = \map h x$ for each $x \in G$.
Since $i$ extends $h$, we have:
- $\map h x = \map i x$ for each $x \in H$.
Since $G \subseteq H$, we in particular have:
- $\map h x = \map i x$ for each $x \in G$.
So, we have:
- $\map g x = \map i x$ for each $x \in G$.
Since $G \subseteq I$, we therefore have:
- $i$ extends $g$.
So, we have:
- $\tuple {G, g} \preceq \tuple {I, i}$
$\Box$
Proof of $(3)$
Suppose that $\tuple {G, g}$ and $\tuple {H, h}$ have:
- $\tuple {G, g} \preceq \tuple {H, h}$
and:
- $\tuple {H, h} \preceq \tuple {G, g}$
We show that:
- $\tuple {G, g} = \tuple {H, h}$
That is:
- $G = H$ and $g = h$.
Since:
- $\tuple {G, g} \preceq \tuple {H, h}$
we have that:
- $G \subseteq H$
and:
- $g$ extends $h$.
Since:
- $\tuple {H, h} \preceq \tuple {G, g}$
we have:
- $H \subseteq G$
and:
- $h$ extends $g$.
So, from the definition of set equality, we have:
- $G = H$
So:
- $h$ is actually a linear functional $G \to \R$.
Since $h$ extends $g$, we have:
- $\map h x = \map g x$ for each $g \in G$.
That is:
- $g = h$
So, we have:
- $\tuple {G, g} = \tuple {H, h}$
$\blacksquare$