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

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

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

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

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

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

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

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

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.