Hahn–Banach Theorem
Extend linear functionals with norm control.
Key idea
Extend linear functionals with norm control.
In depth
A bounded linear functional on a subspace can be extended to the whole space without increasing its norm. It guarantees many functionals exist and supports separation of convex sets.
Worked example
For any non-zero x in a normed space there is a functional f with f(x) = ‖x‖ and ‖f‖ = 1.
Real-world uses
- Extending measurements and constraints to larger spaces in economics and optimisation
Common mistake
Expecting the extension to be unique.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Hahn–Banach Theorem?
Extend linear functionals with norm control. It is a topic at the expert level in Analysis & Topology.
Where is Hahn–Banach Theorem used in real life?
Real-world examples: Extending measurements and constraints to larger spaces in economics and optimisation.
What should I learn before Hahn–Banach Theorem?
Make sure you are comfortable with Bounded Linear Operators, the previous topic in Analysis & Topology.
What is a common mistake with Hahn–Banach Theorem?
Expecting the extension to be unique.
How can I practise Hahn–Banach Theorem?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Hahn–Banach Theorem?
Move on to Open Mapping and Closed Graph next.
Related expert topics in Analysis & Topology
- Axioms of the Real Numbers – ℝ is a complete ordered field.
- Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
- Archimedean Property – For any real x there is an integer n > x.
- Density of Rationals – Between any two reals there is a rational.
- Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
- Limit Theorems for Sequences – Algebra of limits and squeeze theorem.