Some Geometric Consequences of the Hahn-Banach Theorem

Prove the following. If $\mathscr {X}$ is a TVS and $f : \mathscr {X} \to \mathbb {F}$ is a linear functional, then the following statements are equivalent.

1. $f$ is continuous.

2. $f$ is continuous at $0$.

3. $f$ is continuous at some point.

4. $\ker f$ is closed.

5. $x \mapsto \left|f(x)\right|$ is a continuous seminorm. If $\mathscr {X}$ is a LCS and $\mathscr {P}$ is a family of seminorms that defines the topology on $\mathscr {X}$, then the statements above are equivalent to the following:

6. There are $p_1, \dots , p_n$ in $\mathscr {P}$ and positive scalars $\alpha_1, \dots , \alpha_n$ such that $\left|f(x)\right| \le \sum_{k=1}^{n} \alpha_k p_k(x)$ for all $x$.

Recall the following construction (Proposition 3.2): if $\mathscr {X}$ is a TVS and $G$ is an open convex subset of $\mathscr {X}$ containing the origin, and

then $q$ is a non-negative continuous sublinear functional and $G = \left\{ x : q(x) < 1\right\}$.

Now let $p$ be a sublinear functional, $G \equiv \left\{ x : p(x) < 1\right\}$, and define the sublinear functional $q$ for the set $G$ as above. Show that $q(x) = \max (p(x), 0)$ for all $x$ in $\mathscr {X}$.

Let $\mathscr {M} \subseteq \mathscr {X}$, a TVS, and show that the following statements are equivalent:

1. $\mathscr {M}$ is an affine hyperplane;

2. there exists an $x_0$ in $\mathscr {M}$ such that $\mathscr {M} - x_0$ is a hyperplane;

3. there is a non-zero linear function $f : \mathscr {X} \to \mathbb {F}$ and an $\alpha$ in $\mathbb {F}$ such that $\mathscr {M} = \left\{ x \in \mathscr {X} : f(x) = \alpha \right\}$.

Let $\mathscr {X}$ be a real TVS. Show:

If $\mathscr {X}$ is a complex TVS and $f : \mathscr {X} \to \mathbb {C}$ is a nonzero continuous linear function, show that $\mathscr {X} \setminus \ker f$ is connected.

Let $\mathscr {X}$ be a real TVS. Prove the following.

If $f : \mathscr {X} \to \mathbb {R}$ is a continuous $\mathbb {R}$-linear functional and $A$ is an open convex subset of $\mathscr {X}$, then $f(A)$ is an open interval.

Prove the following. If $\mathscr {X}$ is a real LCS and $A \subseteq \mathscr {X}$, then the closed linear span of $A$ is the intersection of all closed hyperplanes containing $A$.

Prove the following. Let $\mathscr {X}$ be a complex LCS and let $A$ and $B$ be two disjoint closed convex subsets of $\mathscr {X}$. If $B$ is compact, then there is a continuous linear functional $f : \mathscr {X} \to \mathbb {C}$, an $\alpha$ in $\mathbb {R}$, and an $\varepsilon > 0$ such that for $a$ in $A$ and $b$ in $B$,

Recall Theorem 3.7: if $\mathscr {X}$ is a real TVS and $A$ and $B$ are disjoint convex sets with $A$ open, then there is a continuous linear functional $f : \mathscr {X} \to \mathbb {R}$ and a real scalar $\alpha$ such that $f(a) < \alpha$ for all $a$ in $A$ and $f(b) \ge \alpha$ for all $b$ in $B$. If $B$ is also open, then $A$ and $B$ are strictly separated.

State and prove a version of this theorem for a complex TVS.

Recall Corollary 3.11: if $\mathscr {X}$ is a real LCS and $A \subseteq \mathscr {X}$, then $\overline{\operatorname {co}}(A)$ is the intersection of the closed half-spaces containing $A$.

State and prove a version of this corollary for a complex LCS.

Recall Corollary 3.12: if $\mathscr {X}$ is a real LCS and $A \subseteq \mathscr {X}$, then the closed linear span of $A$ is the intersection of all closed hyperplanes containing $A$.

State and prove a version of this corollary for a complex LCS.

For $0 < p < 1$, let $L^p(0,1)$ be the collection of equivalence classes of measurable functions $f : (0,1) \to \mathbb {R}$ such that

Given $f, g \in L^p(0,1)$, put $d(f,g) = ((f-g))_p$; this is a translation-invariant metric on $L^p(0,1)$. Prove that $L^p(0,1)$, $0 < p < 1$, is complete with respect to $d$.

Give an example of a TVS $\mathscr {X}$ that is not locally convex and a subspace $\mathscr {Y}$ of $\mathscr {X}$ such that there is a continuous linear functional $f$ on $\mathscr {Y}$ with no continuous extension to $\mathscr {X}$.

Let $\mathscr {X}$ be a real LCS and let $A$ and $B$ be disjoint compact convex subsets of $\mathscr {X}$. Suppose $\mathscr {Y}$ is a subspace of $\mathscr {X}$ and $f_0 : \mathscr {Y} \to \mathbb {R}$ is a continuous linear functional such that $f_0(a) < 0$ for $a$ in $A \cap \mathscr {Y}$ and $f_0(b) > 0$ for $b$ in $B \cap \mathscr {Y}$. Show by an example that it is not always possible to extend $f_0$ to a continuous linear functional on $\mathscr {X}$ such that $f(a) < 0$ for $a$ in $A$ and $f(b) > 0$ for $b$ in $B$.