Duality: Hyperplanes, Functionals, and Hahn–Banach

Overview¶

There are two ways to study a mathematical structure: look at the objects in the space, or look at the functions on the space. In functional analysis, the second viewpoint is remarkably powerful, but it rests on a geometric foundation that is worth making explicit.

In Hilbert spaces, we have orthogonality: the inner product lets us decompose vectors into components, project onto subspaces, and find nearest points. But most Banach spaces have no inner product, so orthogonality is lost. What does survive is hyperplane geometry. Every continuous linear functional $f : X \to \mathbb{R}$ slices the space into parallel hyperplanes (the level sets $f^{-1}(c)$), and its kernel is a codimension-1 subspace. This structure is purely algebraic and topological; it requires no inner product. The kernel determines the functional up to scaling, level sets foliate the space, and closed kernels correspond to continuous functionals. All of this works identically in $\mathbb{R}^3$ and in $L^p(\Omega)$.

Duality is the systematic study of this surviving geometry. The dual space $X^* = \mathcal{L}(X, \mathbb{R})$ collects all bounded linear functionals on $X$, encoding all possible ways to slice the space with hyperplanes. But is $X^*$ rich enough? Could all functionals miss some structure in $X$? The Hahn–Banach Theorem answers this: functionals defined on subspaces can always be extended to the full space, guaranteeing that $X^*$ is large enough to separate points, recover norms, and separate convex sets. From this we get:

• Separation theorems for convex sets, putting the hyperplane geometry to work.

• The Riesz Representation Theorem identifying dual spaces (e.g., $H^* \cong H$ for Hilbert spaces), where orthogonality is recovered as a special case.

• Weak and weak-$*$ convergence, providing compactness where norm convergence fails by testing against functionals.

• Banach–Alaoglu, giving weak-$*$ compactness of the unit ball in $X^*$.

What You Will Learn¶

• How functionals define hyperplanes, and why kernels have codimension 1.

• The geometry of foliations and what distinguishes continuous from discontinuous functionals.

• The Hahn–Banach Theorem and its geometric consequences: separation of convex sets and the sup formula for the norm.

• Dual spaces in practice: Riesz representation for Hilbert spaces, reflexive and non-reflexive spaces.

• Weak and weak-$*$ convergence, and the Banach–Alaoglu compactness theorem.