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.
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
1 · Read
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.
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.
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.
Extend functionals without growth, then separate points by testing against the dual.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.