Hahn–Banach theorem

Key takeaways

Hahn–Banach theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Hahn–Banach theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Hahn–Banach theorem from memory before moving on to harder problems.

Reference excerpt

In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space. The theorem also shows that there are sufficient continuous linear functionals defined on every normed vector space in order to study the dual space. Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the hyperplane separation theorem, and has numerous uses in convex geometry.

History The theorem is named for the mathematicians Hans Hahn and Stefan Banach, who proved it independently in the late 1920s. The special case of the theorem for the space C [ a , b ] {\displaystyle C[a,b]} of continuous functions on an interval was proved earlier (in 1912) by Eduard Helly, and a more general extension theorem, the M. Riesz extension theorem, from which the Hahn–Banach theorem can be derived, was proved in 1923 by Marcel Riesz. The first Hahn–Banach theorem was proved by Eduard Helly in 1912 who showed that certain linear functionals defined on a subspace of a certain type of normed space ( C N {\displaystyle \mathbb {C} ^{\mathbb {N} }} ) had an extension of the same norm. Helly did this through the technique of first proving that a one-dimensional extension exists (where the linear functional has its domain extended by one dimension) and then using induction. In 1927, Hahn defined general Banach spaces and used Helly's technique to prove a norm-preserving version of Hahn–Banach theorem for Banach spaces (where a bounded linear functional on a subspace has a bounded linear extension of the same norm to the whole space). In 1929, Banach, who was unaware of Hahn's result, generalized it by replacing the norm-preserving version with the dominated extension version that uses sublinear functions. Whereas Helly's proof used mathematical induction, Hahn and Banach both used transfinite induction. The Hahn–Banach theorem arose from attempts to solve infinite systems of linear equations. This is needed to solve problems such as the moment problem, whereby given all the potential moments of a function one must determine if a function having these moments exists, and, if so, find it in terms of those moments. Another such problem is the Fourier cosine series problem, whereby given all the potential Fourier cosine coefficients one must determine if a function having those coefficients exists, and, again, find it if so. Riesz and Helly solved the problem for certain classes of spaces (such as L p ( [ 0 , 1 ] ) {\displaystyle L^{p}([0,1])} and C ( [ a , b ] ) {\displaystyle C([a,b])} ) where they discovered that the existence of a solution was equivalent to the existence and continuity of certain linear functionals. In effect, they needed to solve the following problem:

(The vector problem) Given a collection ( f i ) i ∈ I {\displaystyle \left(f_{i}\right)_{i\in I}} of bounded linear functionals on a normed space X {\displaystyle X} and a collection of scalars ( c i ) i ∈ I , {\displaystyle \left(c_{i}\right)_{i\in I},} determine if there is an x ∈ X {\displaystyle x\in X} such that f i ( x ) = c i {\displaystyle f_{i}(x)=c_{i}} for all i ∈ I . {\displaystyle i\in I.}

If X {\displaystyle X} happens to be a reflexive space then to solve the vector problem, it suffices to solve the following dual problem:

(The functional problem) Given a collection ( x i ) i ∈ I {\displaystyle \left(x_{i}\right)_{i\in I}} of vectors in a normed space X {\displaystyle X} and a collection of scalars ( c i ) i ∈ I , {\displaystyle \left(c_{i}\right)_{i\in I},} determine if there is a bounded linear functional f {\displaystyle f} on X {\displaystyle X} such that f ( x i ) = c i {\displaystyle f\left(x_{i}\right)=c_{i}} for all i ∈ I . {\displaystyle i\in I.}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Hahn–Banach theorem

Start with the simplest possible case. Write down what Hahn–Banach theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Hahn–Banach theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Hahn–Banach theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Hahn–Banach theorem

In research: Hahn–Banach theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Hahn–Banach theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Hahn–Banach theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Linear algebra, Linear functionals, Theorems in functional analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Hahn–Banach theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Hahn–Banach theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hahn–Banach theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Hahn–Banach theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hahn–Banach theorem in simple terms?

In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space. The theorem also shows that there are sufficient continuous linear functionals defined on every normed ve…

Why does Hahn–Banach theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Hahn–Banach theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Hahn–Banach theorem.