The Hahn-Banach Theorem and the Dual Space
On a real or complex normed space, a bounded linear functional on a subspace has an extension to the whole space with the same norm. The continuous dual collects bounded linear functionals, which can detect nonzero vectors and separate points from closed subspaces.
What a learner can do afterwards
- 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
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A linear functional f maps vectors to real or complex scalars, with f(au+bv)=af(u)+bf(v). On a normed space, bounded means |f(x)|<=C||x|| for some C. Its norm is sup{|f(x)|: ||x||<=1}. The continuous dual X* is the space of all bounded linear functionals on X.
The kernel is {x: f(x)=0}. For a nonzero functional it is an algebraic hyperplane, meaning codimension one. A bounded functional is continuous, so its kernel is closed: if kernel vectors x_n converge to x, then f(x)=lim f(x_n)=0. Perpendicular arrows describe this kernel in Euclidean space, not in every normed space.
Hahn-Banach extends a bounded linear functional f from any subspace M of a real or complex normed space X to F on X, with F agreeing with f on M and ||F||=||f||. Neither completeness nor closedness of M is required. Any extension has norm at least ||f||, since its unit-ball readings include those on M. An extension with norm at most ||f|| therefore has equal norm.
Let M be a closed subspace and v outside M. The distance d=inf{||v-m||: m in M} is positive. Define g(m+av)=ad on M+span{v}; |g(z)|<=||z||. Hahn-Banach extends g to a bounded functional vanishing on M but nonzero at v. If v instead lies in the closure of M, continuity forces every bounded functional vanishing on M to vanish at v too.
In Euclidean R^n, f(x)=a dot x has norm ||a||. Cauchy-Schwarz gives |f(x)|<=||a||||x||, and a nonzero a attains this bound at x=a/||a||. For f(x,y)=x+2y, the norm is sqrt(5). A different input norm can give a different functional norm.
In finite coordinates, a functional's row times the input column gives the pairing: multiply matching entries and add. For sequences, l1 means sum |x_n| is finite, l infinity means bounded with norm sup |x_n|, and c0 means tending to zero with that norm. The duals of l1 and c0 are l infinity and l1. The pairing is sum a_n x_n; the functional norms are respectively sup |a_n| and sum |a_n|.
For v nonzero, f(av)=a||v|| on span{v} has norm 1. Hahn-Banach extends it to F with F(v)=||v||>0, so bounded functionals detect nonzero vectors. Norm-preserving extensions need not be unique. On R^2 with norm |x|+|y|, both F(x,y)=x and G(x,y)=x+y extend f(x,0)=x with norm 1.
Bounded functionals form the continuous dual. Hahn-Banach preserves their norm on extension and supplies functionals that detect vectors or separate closed subspaces.
Original lesson, questions and flashcards on bounded linear functionals and Hahn-Banach extension, written for this learning dataset.
Original readings, practice questions and flashcards on bounded functionals, separation and sequence-space duals, written for this learning dataset.
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