Dual Spaces & the Hahn–Banach Theorem

The continuous dual $X^*$ is the space of bounded linear functionals $f:X\to\mathbb K$. A dual vector is a measurement: it converts an object into a scalar while respecting linearity and controlling sensitivity. Hahn–Banach says that a bounded measurement defined on a subspace can be extended to the whole space without paying extra norm.

The dual norm and completeness #

For $f\in X^*$,

$$\|f\|=\sup_{\|x\|\le1}|f(x)|.$$

$X^*=\mathcal B(X,\mathbb K)$ is Banach even when $X$ is incomplete, because the scalar codomain is complete. Familiar examples include

$$(\ell^p)^*=\ell^q\quad(1\le p<\infty), \qquad (c_0)^*=\ell^1,$$

with $1/p+1/q=1$. Hilbert-space Riesz representation identifies $H^*$ with $H$, but an arbitrary Banach space need not resemble its dual.

Hahn–Banach extension #

In normed-space form, let $M\subseteq X$ be a linear subspace and $f_0\in M^*$. There exists $f\in X^*$ such that

$$f|_M=f_0, \qquad \|f\|=\|f_0\|.$$

The algebraic theorem extends one dimension at a time while remaining below a sublinear dominating function; Zorn's lemma produces a maximal extension. The complex version follows by extending the real part and reconstructing the complex-linear functional. The theorem guarantees existence, not uniqueness or a constructive algorithm.

Norming functionals and separation of points #

For $x_0\ne0$, define on $M=\mathop{\mathrm{span}}\{x_0\}$

$$f_0(\alpha x_0)=\alpha\|x_0\|.$$

It has norm one. Hahn–Banach extends it to $f\in X^*$ with

$$\|f\|=1,qquad f(x_0)=\|x_0\|.$$

Therefore

$$\|x\|=\sup_{\|f\|\le1}|f(x)|,$$

and continuous functionals separate points. The canonical map $J:X\to X^{**}$, $(Jx)(f)=f(x)$, is consequently an isometry.

Distance to a subspace #

If $M$ is a closed subspace and $x\notin M$, Hahn–Banach produces $f\in X^*$ such that

$$f|_M=0,qquad \|f\|=1,qquad f(x)=\mathop{\mathrm{dist}}(x,M)$$

after choosing the scalar phase appropriately. Functionals that vanish on $M$ form the annihilator

$$M^\perp=\{f\in X^*:f(m)=0\text{ for all }m\in M\}.$$

The dual of the quotient $X/M$ is isometrically identifiable with $M^\perp$. This turns quotient geometry into measurements blind to the removed subspace.

Separating convex sets #

A point outside a closed convex set can be strictly separated from that set by a continuous affine functional under the standard real locally convex/normed space hypotheses. Geometrically, a nonlinear set is replaced by a scalar inequality. This is the engine behind supporting hyperplanes, optimality conditions, and weak closedness of norm-closed convex sets.

Dual-space checklist #

• Use the continuous dual, not the algebraic dual.
• Compute the functional norm in the stated primal norm.
• At $L^1/L^\infty$ endpoints, verify the measure-space assumptions.
• Distinguish annihilators in $X^*$ from orthogonal complements in a Hilbert space.
• Treat Hahn–Banach extensions as generally nonunique.