Hahn-Banach theorem

Theorem, ($\mathbb{R}$ version)

Let $\mathcal{X}$ be an $\mathbb{R}$-vector space. Suppose $q : \mathcal{X} \to \mathbb{R}$ is a quasi-seminorm. Suppose also we are given a linear subspace $\mathcal{Y} \subset \mathcal{X}$ and a linear map $\phi : \mathcal{Y} \to \mathbb{R}$, such that

Then there exists a linear map $\psi: \mathcal{X} \to \mathbb{R}$ such that $\psi |_\mathcal{Y} = \phi$ and $\psi(x) \leq q(x)$ for all $x \in \mathcal{X}$.

Theorem, (normed linear spaces)

#incomplete

Theorem (Helly, Hahn-Banach analytic form)

Let $E$ be a vector space over $\mathbb{R}$.

A functional is a function defined on $E$ or some subspace of $E$ with values in $\mathbb{R}$.

Suppose function $p : E \to \mathbb{R}$ to be a Minkowski functional, i.e. satisfying

• $p(\lambda x) = \lambda p(x) \qquad \forall x \in E \text{ and } \forall \lambda >0$
• $p(x+y) \leq p(x) + p(y) \quad \forall x, y \in E$

Let $G \subset E$ be a linear subspace, and let $g : G \to \mathbb{R}$ be a linear functional such that

• $g(x) \leq p(x) \quad \forall x \in G$

Then under these assumptions, there exists a linear functional $f$ defined on all of $E$ that extends $g$, i.e. $g(x) = f(x) \forall x \in G$, and such that

• $f(x) \leq p(x) \quad \forall x \in E$

Notes

• extension of linear functional defined on a linear subspace of $E$ by a linear functional defined on all of $E$
• This theorem may be proved using Zorn's lemma

References

1. https://www.ucl.ac.uk/~ucahad0/3103_handout_6.pdf
2. https://www.math.ksu.edu/~nagy/real-an/ap-e-h-b.pdf
3. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. New York, NY: Springer New York, 2011, pp. 1-2. doi: 10.1007/978-0-387-70914-7.