Ref:

• Functional Analysis lectures by Casey Rodriguez. Lecture on the Hahn-Banach theorem. Link to the lecture: Link.

Updated: 3/3/26

For a normed space ${ V , }$ let ${ V ^{’} }$ denote ${ V ^{’} = \mathcal{B}(V, K) . }$

Q) Given a general nontrivial normed space ${ V , }$ how large is ${ V ^{’} }$? Is it ${ V ^{’} = \lbrace 0 \rbrace }$?

Note that bounded linear operators ${ V \to K }$ are called “functionals”.

\[{ \underline{\textbf{Zorn's Lemma}} }\]

Def [Partial Order]
Let ${ E }$ be a set. A partial order on ${ E }$ is a relation ${ \preccurlyeq }$ on ${ E }$ such that:

• [Reflexive] For all ${ e \in E , }$ we have ${ e \preccurlyeq e . }$

• [Anti-symmetric] For all ${ e, f \in E , }$ we have ${ e \preccurlyeq f }$ and ${ f \preccurlyeq e }$ implies ${ e = f . }$

• [Transitive] For all ${ e, f, g \in E , }$ we have ${ e \preccurlyeq f }$ and ${ f \preccurlyeq g }$ implies ${ e \preccurlyeq g . }$

Def [Upper Bound]
Let ${ E }$ be a set with a partial order ${ \preccurlyeq . }$ Let ${ D \subseteq E . }$
An upper bound of ${ D }$ is an element ${ e \in E }$ such that

\[{\forall d \in D, \quad d \preccurlyeq e . }\]

Informally, ${ e }$ dominates every element of ${ D . }$

Def [Maximal Element]
Let ${ E }$ be a set with a partial order ${ \preccurlyeq . }$
A maximal element of ${ E }$ is an element ${ e \in E }$ such that

\[{ (f \in E \, \, \text{ and } \, \, e \preccurlyeq f) \implies e = f . }\]

Informally, the only element of ${ E }$ which dominates ${ e }$ is ${ e . }$

Eg [A maximal element which isn’t an upper bound]
Link to Stackexchange post: Link. Consider ${ \lbrace 1, \ldots, 5 \rbrace }$ with equality.

Eg [An upper bound which isn’t a maximal element]
Consider ${ (0, 1) }$ and ${ 2 . }$

Def [Chain]
Let ${ (E, \preccurlyeq) }$ be a poset (partially ordered set). Let ${ C \subseteq E . }$
We say ${ C }$ is a chain in ${ E }$ if

\[{ \forall e, f \in C , \quad e \preccurlyeq f \, \, \text{ or } \, \, f \preccurlyeq e . }\]

Informally, any two elements of ${ C }$ are comparable.

Lemma [Zorn’s Lemma]
Let ${ E }$ be a non empty poset. Suppose every chain has an upper bound.
Then ${ E }$ has a maximal element.

To study later: Zorn’s lemma: Intuition and Applications.

Def [Hamel Basis]
Let ${ V }$ be a vector space. A Hamel basis ${ H \subseteq V }$ is a linearly independent set where every element of ${ V }$ is a finite linear combination of elements of ${ H . }$

It turns out every vector space has a Hamel basis.

Thm [Hamel Basis]
Let ${ V }$ be a vector space. Then ${ V }$ has a Hamel basis.

Pf: Let

\[{ E = \lbrace \text{lin. indep. subsets of V} \rbrace }\]

with inclusion.

We will try to apply Zorn.

Let ${ C }$ be a chain in ${ E . }$ We are looking for an upper bound of ${ C . }$

Define

\[{ c = \bigcup _{e \in C} e . }\]

We will show ${ c \in E .}$ Then ${ c }$ is an upper bound for ${ C . }$

Let ${ v _1, \ldots, v _n \in c. }$ There exist ${ e _1, \ldots, e _n \in C }$ such that each ${ v _j \in e _j . }$ Since we can compare elements in ${ C , }$ we can order ${ e _1, \ldots, e _n . }$ Hence there exists a ${ J }$ such that ${ v _1, \ldots, v _n \in e _J . }$ Hence ${ (v _1, \ldots, v _n) }$ are linearly independent.

Hence

\[{ \text{c is an upper bound of C} }\]

as needed.

Hence by Zorn’s lemma, ${ E }$ has a maximal element ${ H . }$

We will show ${ H }$ spans ${ V . }$

Suppose ${ H }$ doesnt span ${ V . }$ Then there exists ${ v \in V }$ such that ${ v }$ cannot be written as a finite linear combination of elements of ${ H . }$ Then ${ H \cup \lbrace v \rbrace }$ is linearly independent, contradicting maximality of ${ H . }$

Hence ${ H }$ spans ${ V . }$ Hence ${ H }$ is a Hamel basis, as needed.

We will now prove the Hahn-Banach Theorem.

\[{ \underline{\textbf{Hahn-Banach Theorem}} }\]

Thm:
Let ${ V }$ be a normed space. Let ${ M \subseteq V }$ be a subspace.

Let ${ u : M \to \mathbb{C} }$ be bounded linear such that

\[{ \forall t \in M, \quad \vert u(t) \vert \leq C \lVert t \rVert . }\]

Then there exists bounded linear ${ U \in V ^{‘} }$ such that ${ U \vert _M = u }$ and

\[{ \forall t \in V, \quad \vert U(t) \vert \leq C \lVert t \rVert . }\]

Here we say “${ U }$ is a continuous extension of ${ u }$” for brevity.

To study later: Hahn-Banach Theorem: Intuition and Applications.