Functional Analysis 25 | Hahn–Banach Theorem [dark version]

The Hahn–Banach theorem for normed spaces guarantees that continuous linear functionals defined on a subspace can be extended to the whole space without increasing their norm—an idea that turns the dual space into a powerful tool for probing the original space. In its extension form, start with a continuous linear functional on a subspace U of a normed space X. Hahn–Banach produces a continuous linear functional on all of X that agrees with the original on U and preserves the operator norm, making the dual space “large enough” to carry detailed information about X.

A key application uses this extension principle to build norm-attaining functionals. For any nonzero vector x in X, one considers the one-dimensional subspace U spanned by x. On U, define a functional U′ by sending λx to λ‖x‖ (with values in the underlying field F, real or complex). This functional is continuous and has norm 1. Hahn–Banach then extends it to a functional X′ on the entire space X such that X′(x)=‖x‖ and ‖X′‖=1. The payoff is immediate: the dual space separates points. Given two distinct vectors x1 and x2, their difference x=x2−x1 is nonzero, so the construction yields a functional X′ with X′(x)=‖x‖≠0. Linearity then forces X′(x2)−X′(x1)≠0, meaning X′(x1) and X′(x2) cannot be equal. In practical terms, no two different points in X look identical to all continuous linear functionals.

That separation result feeds into a second, more quantitative identity: the norm of a vector can be recovered from the dual. The norm satisfies ‖x‖ = sup { |X′(x)| : X′ ∈ X′, ‖X′‖=1 }. One direction follows from the definition of operator norm: for any functional X′ with ‖X′‖=1, the value |X′(x)| cannot exceed ‖x‖. The other direction uses the earlier Hahn–Banach construction that produces a functional of norm 1 attaining the value ‖x‖ on x, turning the supremum into a maximum.

The final application addresses how closed subspaces can be “annihilated” by nontrivial functionals. If U is a closed subspace of X and x is a vector not in U, there exists a continuous linear functional X′ on X that vanishes on every element of U but not on x. The argument passes through the quotient space X/U, where vectors differing by an element of U become equivalent. In X/U, the equivalence class of x is nonzero precisely because x∉U. Hahn–Banach applied on the quotient yields a functional y′ on X/U that is zero at the zero class but nonzero at the class of x. Pulling y′ back to X defines X′ so that X′|U = 0 while X′(x) ≠ 0. This is a standard mechanism for constructing separating/annihilating functionals in functional analysis, especially when closedness is needed to ensure the quotient carries a well-behaved norm.