Hahn–Banach Theorem

The Hahn–Banach theorem is a fundamental result in functional analysis asserting that a linear functional defined on a subspace can be extended to the entire space without increasing its prescribed bound. Its most familiar form preserves the norm of a bounded linear functional. Related geometric forms separate convex sets by hyperplanes. Together, these results establish that continuous linear functionals provide sufficiently many scalar measurements to distinguish vectors and describe their norms. (ocw.mit.edu)

Norm-preserving extension

Let $X$ be a normed vector space over $\mathbb F=\mathbb R$ or $\mathbb C$, and let $M\subseteq X$ be a linear subspace. Suppose $f:M\to\mathbb F$ is linear and bounded, with norm

Then there exists a bounded linear functional $F:X\to\mathbb F$ such that

Neither completeness of $X$ nor closedness of $M$ is required. Thus the theorem applies to normed spaces generally, not only to Banach spaces. (ocw.mit.edu)

The conclusion is stronger than merely extending a linear map algebraically: it preserves the smallest constant controlling its size. If $|f(x)|\leq C\|x\|$ on $M$, the extension can satisfy the same inequality throughout $X$. (ocw.mit.edu)

Dominated extension

A more general real formulation uses a sublinear function $p:X\to\mathbb R$, where $X$ is a real vector space. Sublinearity means

If a linear functional $f:M\to\mathbb R$ satisfies

then it has a linear extension $F:X\to\mathbb R$ satisfying

This formulation is algebraic: no topology or continuity is assumed. Unlike a norm, a sublinear function need not be symmetric or nonnegative. (bpb-us-e1.wpmucdn.com)

For a seminorm $p$, the theorem also has a real or complex version:

Taking $p(x)=\|f\|\|x\|$ gives norm-preserving extension: the bound yields $\|F\|\leq\|f\|$, while restriction to $M$ yields the reverse inequality. (web.ma.utexas.edu)

Proof mechanism

The essential step extends a real functional by one dimension. Choose $z\notin M$ and write

To maintain $g\leq p$, it is enough to choose $a$ satisfying

This interval is nonempty because, for $u,v\in M$,

Positive homogeneity then handles arbitrary positive and negative coefficients $t$. (loss.math.gatech.edu)

To reach the entire space, order all dominated extensions by extension of their domains. Every chain has an upper bound obtained by taking its union. Zorn’s lemma gives a maximal extension; the one-dimensional step shows that its domain cannot be a proper subspace. (ocw.mit.edu)

The complex case is reduced to the real case. Extend $\operatorname{Re}f$ to a real-linear functional $u$ on the underlying real space, then define

This is complex-linear and extends $f$. A rotation by a complex scalar of modulus one converts the bound on $\operatorname{Re}F$ into the required bound on $|F|$. (web.ma.utexas.edu)

Geometric separation

Geometric Hahn–Banach theorems translate extension into statements about convex sets. In a real locally convex topological vector space, two nonempty disjoint convex sets $A,B$, one of which is open, admit a nonzero continuous linear functional $\ell$ and a real number $\alpha$ such that

The set $\{x:\ell(x)=\alpha\}$ is a separating hyperplane. For complex spaces, separation is expressed using the real part of a complex-linear functional. (arxiv.org)

A useful stronger case separates a point $x_0$ from a nonempty closed convex set $C$ not containing it:

The hypotheses matter: arbitrary disjoint convex sets need not admit a strict separating gap. One bridge between geometry and extension is the Minkowski functional

associated with a suitable convex neighborhood $U$ of zero. Its sublinearity allows the extension theorem to construct separating functionals. (bpb-us-e1.wpmucdn.com)

Consequences for duality

Let $X^*$ denote the continuous dual space of a normed space $X$. For each nonzero $x\in X$, Hahn–Banach provides $f\in X^*$ with

To obtain it, define $f_0(\lambda x)=\lambda\|x\|$ on the span of $x$, then extend. Consequently,

In particular, continuous linear functionals separate distinct points. (ocw.mit.edu)

The canonical map into the second dual,

is therefore a linear isometry. Hahn–Banach does not assert that $J$ is onto; surjectivity is the additional property called reflexivity. (ocw.mit.edu)

These facts explain why the theorem is central to duality: vector norms can be recovered from scalar-valued tests, even when no inner-product representation of those tests is available. (ocw.mit.edu)

Nonuniqueness and limitations

Extensions need not be unique. For an explicit example, equip $\mathbb R^2$ with

On the horizontal axis, set $f(x,0)=x$. Every

extends $f$ and has norm one: $|F_b(x,y)|\leq|x|+|y|$, with equality at $(1,0)$.

The theorem concerns scalar-valued functionals. An unrestricted norm-preserving extension theorem for operators taking values in arbitrary Banach spaces is false; vector-valued extension requires additional conditions. Moreover, the maximality argument proves existence without generally supplying an explicit formula or computational procedure. (sciencedirect.com)

Zorn’s lemma is equivalent to the axiom of choice, but using it in the proof does not make Hahn–Banach equivalent to full choice. The general theorem follows from weaker choice principles, including the ultrafilter theorem. (sciencedirect.com)

Historical development

The theorem is named after Hans Hahn and Stefan Banach, whose independent work appeared in 1927 and 1929, respectively. Earlier results of Frigyes Riesz and Eduard Helly anticipated parts of the extension principle. The development from extension results to geometric separation and duality helped establish the framework of modern functional analysis. (sciencedirect.com)

References

1. 102 S2021 Lecture 5. Zorn's Lemma and the Hahn-Banach Theoremocw.mit.edu
2. 102 S2021 Lecture 6. The Double Dual and the Outer Measure of a Subset of Real Numbersocw.mit.edu
3. Lectures on Analysisbpb-us-e1.wpmucdn.com
4. Proof of the Hahn-Banach Theoremloss.math.gatech.edu
5. Methods of Applied Mathematicsweb.ma.utexas.edu
6. The Hahn-Banach Theorem: a proof of the equivalence between the analitic and geometric versionsarxiv.org