I just started reading functional analysis and came across the Hahn-Banach theorem. Everywhere I looked stated this was very important and it is because it allows us to enlarge a dual space $X^+.$ I keep wondering to myself, so what? (sorry for sounding rude). If we can enlarge the dual space of $X^+$, then what? Why are the extensions of linear functionals so important? I mean for a subspace $Z \subset X$, is there something important we need to know about $X-Z$ that we need to extend a linear functional $f: Z \to \mathbb{F}$ on?

Is there an application or simple example of this outside of functional analysis that motivates the idea? Even something simple in Differential Equations would be insightful or a simple problem in Linear Algebra (actually this is probably dumb since I would imagine you would just extend the basis, so we have to give an example for a infinite dim space; maybe showing an analogue of Hahn-Banach in this setting would be neat and somewhat convincing)

Right now I just can't be bothered reading the full proof of something that I don't see the importance in and only to forget about it later...

Thank you.

  • 1
    $\begingroup$ The Hahn-Banach theorem implies that the weak topologies of Banach spaces (for example) are Hausdorff. This leads to the notion of weak (distributional) solutions in partial differential equations, which is fundamental in many areas of the subject. $\endgroup$ – fourierwho Dec 15 '17 at 22:21

The goal of Functional Analysis is to study Topological Vector Spaces (TVSs), which are vector spaces endowed with topologies making their algebraic operations continuous. The reason why it's called Functional Analysis is because linear functionals (which are maps from the vector space into its scalar field) are the tools that you use to analyze these TVSs. Why use linear functionals? Because they're maps that take your complicated infinite-dimensional TVS, say $X$, and map it into $\mathbb{R}$, thereby allowing you to "see" one dimension of $X$ at a time and also allowing you to apply all the tools for dealing with $\mathbb{R}$ (e.g. real analysis) to this image of this dimension of $X$. This is important for the study of the Calculus of Variations, for instance. Using linear functionals you can, for example, determine whether or not two things (e.g. points or maps) are equal (this important but under-appreciated property is the motivation for the definition of "a dual space that separates points") or determine whether or not a set is closed or compact in the weak topology (e.g. the Banach-Alaoglu Theorem), which then gives you information about the set in the original TVS topology. Linear functionals play a big role in, for instance, the theory of Nuclear spaces, which has applications to Quantum Mechanics. Distributions (A.K.A. generalized functions) are by definition continuous linear functionals on a specific TVS and distributions allow you to extend differential equations (e.g. the wave equation) to situations where not everything is differentiable (e.g. the wave equation representing a guitar string that is initially shaped like a V). Because of this, distributions appear in many fields. These linear functionals are also important since they make rigorous some important and seemingly impossible concepts such as the well-known Dirac delta function, whose integral over $\mathbb{R}^n$ is $1$ despite it being $0$ everywhere except for at a single point.

Often, for a specific application, you find some subspace of a TVS with some desirable properties (usually with respect to some specific map) and you want to somehow extend your map to the whole space (or somehow relate this desirable property back to the whole space). This is where the Hahn-Banach Theorem comes in since you generally proceed to define some continuous linear functional with some nice property on the subspace and then extend it to some continuous linear functional on the original space.

Also, a TVS topology induces a canonical set of continuous linear functionals (the continuous dual space). However, in general (i.e. when working in infinite dimensions), many different TVS topologies can induce the same set of continuous linear functionals. It's not uncommon to have several different TVS topologies on the same vector space (e.g. see Wikipedia's article Operator topologies:List of topologies on B(H)). Continuous linear functionals thus allow you to "transfer information" between two TVS topologies that have the same (or at least somehow related) continuous dual spaces and the main tool for constructing new continuous linear functionals is the Hahn-Banach Theorem.


I personally really like the following application to a very natural and basic question in measure theory:

There exists a finitely additive measure $m:\mathcal{P}(\mathbb{R}) \to \mathbb{R}_{\geq 0}$ defined on all subsets of the non-negative real line that is translation invariant (i.e. $m(x+A) = m(A)$) that agrees with the usual Lebesgue measure on measurable sets. The proof relies on the Hahn Banach theorem and is done in an ingenious way, one can find it in Stein and Shakarchi's book on functional analysis.

  • $\begingroup$ This isn't an application of Functional Analysis to outside Functional Analysis, but one of my favorite proofs in Functional Analysis is the proof Lomonosolv's Invariant Subspace Theorem (as found in Rudin's "Functional Analysis"). It's short (about a page or two), contains several ingenious tricks/insights one after another, and (according to one of my former instructors) made stacks of publications on the Invariant Subspace Problem redundant. $\endgroup$ – Matthew K. Dec 1 '19 at 6:11

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