##  [Hahn–Banach Theorem](/hahn-banach-theorem-0) 

 Definition

A collection of extension theorems asserting that a bounded linear functional defined on a subspace of a normed vector space can be extended to the whole space without increasing its norm; more generally, linear functionals dominated by a sublinear functional can be extended while preserving domination.

 

 

 

 

 

 





## Principle

Principle

Linear functionals can be prolonged from smaller domains to larger ones while preserving norm or domination constraints; the argument exploits convexity and Zorn's lemma (or equivalent forms of choice) in general abstract formulations.

 

 

 

 

 





## Demonstration

Demonstration

Concrete usage: given a continuous linear functional defined on a one-dimensional subspace spanned by x in a normed space, Hahn–Banach provides an extension to the whole space keeping the same norm; this is used to separate a point from a closed subspace by a continuous functional.

 

 

 

 

## Misapplication

Misapplication

Assuming Hahn–Banach yields unique extensions, constructive formulas, or that it produces norm-attaining functionals in every Banach space are misuses. Relying on it in frameworks rejecting choice also misapplies its standard proof ingredients.

 

 

 

 

 





## Consequence

Consequence

Generates nontrivial continuous linear functionals on many spaces, underlies separation theorems, duality theory, and weak topologies; it is a foundational tool in functional analysis and convex analysis.

 

 

 

 

## Reversal

Reversal

If no extension principle holds, dual spaces shrink and separation results fail; the reversal highlights how crucial extension is to the richness of continuous duals and geometric separation in infinite dimensions.

 

 

 

 

 





## Boundary

Boundary

Applies to linear functionals and sublinear dominations on vector spaces over R or C; different versions have different hypotheses (normed spaces, locally convex spaces). It does not extend nonlinear operators or guarantee algebraic uniqueness or explicit construction in all contexts.

 

 

 

 

 





## Semantic Tension

Semantic Tension

There is tension between the algebraic form (which can be proven without topology) and analytic norm-preserving versions (which interact with topology and choice), and between existence of extensions and their nonconstructive nature.

 

 

 

 

 





## Synthesis

Synthesis

Hahn–Banach encapsulates the principle that bounded linear information given on a subspace can be consistently extended to the whole space without enlarging its size, providing the analytic and geometric backbone of duality and separation in functional analysis.