In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space. The theorem also shows that there are sufficient continuous linear functionals defined on every normed vector space in order to study the dual space .

THE HAHN-BANACH THEOREM A subspace W of V has codimension 1 if there is a vector x 2 V nW such that W +Rx = V. This is equivalent to saying that the quotient space V=W has dimension 1. A hyperplane is a set of the form W +x where W is any codimension one subspace and x is any vector. Let W be a codimension 1 subspace of V, and v any vector ...

There are several versions of the Hahn-Banach Theorem. Theorem E.1 (Hahn-Banach, R-version). Let X be an R-vector space. Suppose q: X → R is a quasi-seminorm. Suppose also we are given a linear subspace Y ⊂ X and a linear map φ: Y → R, such that φ(y) ≤ q(y), for all y∈ Y. Then there exists a linear map ψ: X → R such that (i) ψ Y ...

The proof of the Hahn–Banach theorem has two parts: First, we show that ℓ can be extended (without increasing its norm) from M to a subspace one dimension larger : that is, to any subspace M 1 = span{M,x 1 } = M +Rx 1 spanned by M and a vector x 1 ∈ X \M.

It will begin with some basic separation results in Rn, such as the Hyperplane Separation Theorem of Hermann Minkowski, and then it will focus on and prove the extension of this theorem into normed vector spaces, known as the Hahn-Banach Separation Theorem.

THEOREM 2.1. (Hahn-Banach Theorem, Positive Cone Version) Let P be a cone in a real vector space X, and let Y be a subspace of X having the property that for each x ∈ X there exists a y ∈ Y such that y ≥ x; i.e., y−x ∈ P. Suppose f is a positive linear functional on Y, i.e., f(y) ≥ 0 if y ∈ P ∩Y.

Can we construct a linear functional f 2 X such that f = f on Z? The Hahn{Banach Theorem gives an a rmative answer to these ques-tions. It provides a poverful tool for studying properties of normed spaces using linear functionals. The proof of the Hahn-Banach theorem is using an inductive argument.

Paul Garrett: Hahn-Banach theorems (May 17, 2019) [3.0.1] Theorem: For a non-empty convex open subset Xof a locally convex topological vectorspace V, and a non-empty convex set Y in V with X\Y = ˚, there is a continuous real-linear real-valued functional on V and a constant cso that (x) < c (y) (for all x2Xand y2Y) Proof: Fix x o2Xand y o2Y ...

In mathematics, specifically in functional analysis and Hilbert space theory, vector-valued Hahn–Banach theorems are generalizations of the Hahn–Banach theorems from linear functionals (which are always valued in the real numbers or the complex numbers) to linear operators valued in topological vector spaces (TVSs).

Consider a subspace $M$ of a complex normed vector space $(X,\|\cdot\|)$, $p:X\to\mathbb{R}$ sublinear and $f:M\to\mathbb{C}$ linear such that $|f(x)|\le p(x)$ $\forall x\in M$. We want to prove that