A family of extension results that guarantee a bounded linear functional defined on a linear subspace of a real or complex normed vector space can be extended to the whole space without increasing its norm; equivalent algebraic forms cover extension dominated by a sublinear functional.

Principle

If a linear functional on a subspace is dominated by a continuous sublinear functional (or is bounded), then there exists an extension to the entire space preserving linearity and the domination (hence the norm).

Demonstration

In a normed space X, given a continuous linear functional f on a subspace Y with ||f|| = c, Hahn-Banach produces F on X with F|_Y = f and ||F|| = c; geometrically this yields linear functionals that separate points from closed convex sets.

Misapplication

Assuming Hahn-Banach provides constructive, canonical extensions or that it extends arbitrary nonlinear maps; also misusing it to extend bounded linear operators between spaces (it applies to functionals, not general operators) without further conditions.

Consequence

A rich dual space: existence of nontrivial continuous linear functionals, hyperplane separation of convex sets, identification and study of duals and reflexivity, and many functional-analytic techniques that depend on abundance of linear functionals.

Reversal

The inverse situation would be a theory where bounded linear functionals on subspaces cannot in general be extended to the whole space without changing norm, yielding few separating functionals and weaker dual-space structure.

Boundary

Applies to real or complex vector spaces with sublinear dominating functionals or normed-space settings; some proofs use the axiom of choice (Zorn's lemma), and constructive extensions may fail; it does not extend arbitrary nonlinear maps or general bounded operators.

Semantic Tension

Tension exists between algebraic (Hahn-Banach for linear forms and sublinear functionals) and topological formulations (norm-preserving extensions); users sometimes conflate extension of functionals with extension of operators or with explicit constructive formulas.

Synthesis

Hahn-Banach unites algebraic domination by sublinear functionals and topological boundedness to produce norm-preserving extensions of linear functionals, enabling separation and dual-space constructions that are central to functional analysis.