---
title: "The Hahn-Banach Theorem and the Dual Space"
description: "A bounded linear functional defined on a subspace extends to the whole space without growing in norm. The dual space these functionals populate is what makes it possible to argue about a vector by tes"
canonical: https://lightmysky.com/learn/mathematics/the-hahn-banach-theorem-and-the-dual-space-mt_pJuHFxkvjS
source: https://lightmysky.com/learn/mathematics/the-hahn-banach-theorem-and-the-dual-space-mt_pJuHFxkvjS.md
retrieved: 2026-09-12
---

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# The Hahn-Banach Theorem and the Dual Space

A bounded linear functional defined on a subspace extends to the whole space without growing in norm. The dual space these functionals populate is what makes it possible to argue about a vector by testing it against everything.

Subject: Mathematics · Area: Calculus & Analysis · Ages 23 to 24
Page: https://lightmysky.com/learn/mathematics/the-hahn-banach-theorem-and-the-dual-space-mt_pJuHFxkvjS

## Ready when they can

- State the extension theorem and say what quantity is preserved
- Use a functional to separate a point from a closed subspace
- Identify the dual of a familiar sequence space or function space

## Lesson: Measure vectors by testing them everywhere

A linear functional collapses vectors to numbers along one fixed direction, like a dot product with a hidden arrow. Its kernel is the hyperplane perpendicular to that arrow. Thinking of flattening a grid onto a single axis keeps the algebra honest: the norm is the steepest slope of the flattening.

The extension theorem says a bounded functional built on a subspace always stretches to the whole space without growing in norm. The preserved quantity is the norm itself: the extended measuring device is exactly as strong as the original, no stronger. New axes inherit the flattening consistently.

**Example.** Functionals separate points from closed subspaces. Given a nonzero vector outside a closed subspace, some bounded functional vanishes on the subspace yet reads nonzero on the vector. Arguing about a vector by testing it against everything is what the dual space makes possible.

**Tip.** Read duals through row vectors acting on column vectors: a one-by-n row eating an n-by-one column is the dual pairing in miniature. In finite dimensions every functional is dotting with some arrow, and the arrow length is the norm. Infinite-dimensional duals scale up the same picture.

**Recap.** Extend functionals without growth, then separate points by testing against the dual.

## Practice

18 questions on this page, each with its working shown.

## Needs first

- [Baire Category and the Uniform Boundedness Principle](https://lightmysky.com/learn/mathematics/baire-category-and-the-uniform-boundedness-principle-mt_11Rd73trAX)
- [Bounded Linear Operators and the Operator Norm](https://lightmysky.com/learn/mathematics/bounded-linear-operators-and-the-operator-norm-mt_Qi_mbSNMMx)
- [Vector Spaces and Subspaces](https://lightmysky.com/learn/mathematics/vector-spaces-and-subspaces-mt_Zrddx-E6_n)

## Opens up

- [Hilbert Spaces and Orthogonal Projection in Infinite Dimensions](https://lightmysky.com/learn/mathematics/hilbert-spaces-and-orthogonal-projection-in-infinite-dimensions-mt_xmKq8SAgvy)