Duality

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.

The dual space $X^* = \mathcal{L}(X, \mathbb{R})$ collects all bounded linear functionals on $X$. Functionals are “measurements”—a scale measures weight, a CT scanner measures line integrals—and $X^*$ captures all possible linear measurements of elements in $X$.

The central result is the Hahn–Banach Theorem, which guarantees that functionals defined on subspaces can always be extended to the full space. This leads to:

• Separation theorems for convex sets.

• The Riesz Representation Theorem identifying dual spaces (e.g., $H^* \cong H$ for Hilbert spaces).

• Weak and weak- convergence*, which provide compactness where norm convergence fails.

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

What You Will Learn¶

• Dual spaces, isomorphisms, and isometries.

• The Hahn–Banach Theorem and its geometric consequences.

• Riesz representation for Hilbert spaces.

• Reflexive spaces and weak convergence.

• Weak-* convergence and the Banach–Alaoglu Theorem.