Hahn - Banach theorems

The first point here is that convex sets can be separated by linear functionals. Second, continuous linear functionals on subspaces of a locally convex topological vectorspace have continuous extensions to the whole space. Proofs are for real vectorspaces. The complex versions are corollaries. A crucial corollary is that on locally convex topological vectorspaces continuous linear functionals separate points, meaning that for x = y there is a continuous linear functional λ so that λ(x) = λ(y). This separation property is essential in applications. Thus, the hypothesis of local convexity is likewise essential. Let k be R or C with usual absolute value, and V a k-vectorspace, without assumptions about topology on V for the moment. A k-linear k-valued function on V is a linear functional. When V has a topology it makes sense to speak of continuity of functionals. The space of all continuous linear functionals on V is denoted V * , suppressing reference to k. A linear functional λ on V is bounded if there is a neighborhood U of 0 in V and constant c such that |λx| ≤ c for x ∈ U. The following proposition is the general analogue of the assertion for Banach spaces, in which the boundedness has a different sense. [1.0.1] Proposition: The following three conditions on a linear functional λ on a topological vectorspace V over k are equivalent: • λ is continuous. • λ is continuous at 0. • λ is bounded. Proof: The first implies the second. Assume the second. Given ε > 0, there is a neighborhood U of 0 such that |λ| is bounded by ε on U. This proves boundedness in the topological vector space sense. Finally, suppose that |λx| ≤ c on a neighborhood U of 0. Then, given x ∈ V and given ε > 0, take y ∈ x + ε · U With x − y = ε · u with u ∈ U , |λx − λy| = ε|λu| ≤ ε · c Rewriting the argument replacing ε by a suitable multiple gives the desired result. In this section, vectorspaces are real.