Review of Linear Forms and the Hahn-Banach Theorem

# Review of Linear Forms and the Hahn-Banach Theorem

• We defined the Algebraic Dual to be $E^*$, the set of all linear forms on $E$, and proved the existence of many nonzero linear forms on $E$. In particular,

If $E$ is a vector space then for each $a \neq o$ there exists a $f \in E^*$ such that $f(a) \neq 0$.

• We also observed that if $E$ is a topological vector space then every $f \in E^*$ is an open map.
 Criteria for a Linear Form to be Continuous (1) $f$ is a continuous linear form. (2) There exists a neighbourhood $U$ of the origin for which $f(U)$ is a bounded set of real or complex numbers. (3) $f^{-1}(0)$ is a closed set (i.e., the null space is closed).
• We also proved the following basic properties regarding continuity when $E$ is further a locally convex topological vector space:
 Properties of Continuity in a Locally Convex Topological Vector Space (1) If $f$ is a linear form that is dominated by a continuous seminorm $p$ then $f$ is continuous. (2) If $f$ is a continuous linear form then $|f|$ is a continuous seminorm.
• On the Hyperplanes of a Vector Space page we defined a Hyperlane of a vector space $E$ to be a maximal and proper subspace $H$ of $E$, or equivalently, a subspace $H$ of $E$ for which $\mathrm{codim}(H) = 1$. We have the following characterizations of hyperplanes in topological vector spaces and locally convex topological vector spaces:
 Characiterizations of Hyperplanes in Topological Vector Spaces and Locally Convex Topological Vector Spaces (1) If $E$ is a topological vector space then $H$ is a hyperplane of $E$ if and only if there exists a nonzero linear form $f$ for which $f^{-1}(0) = H$. (2) If $E$ is a locally convex topological vector space then every hyperplane of $E$ is either closed or dense.
• We then turned our attention to The Hahn-Banach Theorem which is discussed on the following pages:
• The Hahn-Banach Theorem states:

(Hahn-Banach): If $E$ is a vector space, $M$ a subspace of $E$, $f$ a linear form on $M$, and $p$ a seminorm on $E$ for which $|f(x)| \leq p(x)$ for all $x \in M$, then there exists a linear form $f_1$ on $E$ which extends $f$ and such that $|f_1(x)| \leq p(x)$ for all $x \in E$.

• This we proven in multiple steps. We first proved:

If $E$ is a real locally convex topological vector space and if $H$ is subspace that does not intersect some open set $A$ in $E$, then either $H$ is a hyperplane OR there exists a point $x \not \in H$ for which $\mathrm{span} (H \cup \{ x \})$ still does not intersect $A$.

• We then proved that if $H$ is a real hyperplane of $E$, then $H \cap (iH)$ is a complex hyperplane of $E$. Then we showed that:

If $E$ is a locally convex topological vector and if $M$ is a subspace of $E$ that does not intersect some open and convex set $A$ then there must exist a closed hyperplane $H$ of $E$ that contains $M$ and that still does not intersect $A$.

• As a corollary, we observe that for a locally convex topological vector space $E$, a subspace $M$ is the intersection of all closed hyperplanes containing $M$, and from these results, we proved the Hahn-Banach theorem.
 Corollaries to the Hahn-Banach Theorem (1) If $E$ is a locally convex topological vector space and $M$ is a subspace of $E$ then every continuous linear form on $M$ can be extended to a continuous linear form on $E$. (2) If $E$ is a vector space then for each seminorm $p$ and each $a \in E$ there exists a linear form $f$ such that $|f(x)| \leq p(x)$ for all $x \in E$ and $f(a) = p(a)$. (3) If $E$ is a Hausdorff locally convex topological vector space and if $f(a) = 0$ for all $f \in E'$ then $a = o$.

(Hahn-Banach Separation): If $E$ is a locally convex topological vector space then every pair of disjoint convex sets $A$ and $B$ for which at least one of them is open, can be separated by a continuous linear form $f$, i.e., there exists an $f \in E'$ such that $f(A) \cap f(B) = \emptyset$.

• We summarize the corollaries to the Hahn-Banach separation theorem below.
 Corollaries to the Hahn-Banach Separation Theorem (1) If $E$ is a locally convex topological vector space and $B$ is convex with $a \not \in \overline{B}$ then there exists $f \in E'$ such that $f(a) \not \in \overline{f(B)}$. (2) If $E$ is a locally convex topological vector space and $B$ is absolutely convex with $a \not \in \overline{B}$ then there exists $f \in E'$ such that $|f(x)| \leq 1$ on $B$ and $f(a) > 1$. (3) If $E$ is a real locally convex topological vector space and $A$, $B$ is a pair of disjoint convex sets with $A$ open, then there exists $f \in E'$ and a constant $\alpha > 0$ such that $f(x) > \alpha$ on $A$ and $f(x) \leq \beta$ on $B$.