An introduction to the Hahn-Banach extension theorem: Part I

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In this series of posts (intended for mathematics undergraduate students in their 3rd or 4th year) I am going to focus on the Hahn-Banach extension theorem for bounded linear functionals on normed spaces. I may also say something about more general versions in terms of sublinear functions. 

Let $\F$ be $\R$ or $\C$, and let $(E,\jnorm)$ be a normed space over $\F$. We denote by $E^*$ the set of continuous (bounded) linear functionals on $E$, that is, the set of continuous linear maps from $(E,\jnorm)$ to $(\F,|\cdot|).$

Since $(\F,|\cdot|)$ is complete, it follows that $E^*$ is always complete when given the operator norm $\jnorm_\op$ defined, for $\phi\in E^*$, by

\[\|\phi\|_\op = \sup\{|\phi(x)|:x \in E, \|x\|\leq 1\,\}\,.\]

Now let $F$ be a linear subspace of $E,$ and give $F$ the norm obtained by restricting $\jnorm$ to $F.$ We still denote this restricted norm by  $\jnorm$. We now have another dual space $F^*,$ along with a restriction map $R:E^*\to F^*$ defined by $R(\phi)=\phi|_F\,.$

In fact $R(\phi)=\phi \circ \iota$, where $\iota$ is the (isometric) inclusion map from $F$ into $E$, defined by $\iota(x)=x$ for all $x \in F.$

In terms of dual maps, we have $R=\iota^*$.

It is clear that $R$ is linear, and that $\|R\|_\op \leq 1.$ More explicitly, for all $\phi \in E^*$,

\[\|\phi|_F\|_\op \leq \|\phi\|_\op\,.\]

Let $\phi \in E^*,$ and let $\psi \in F^*$. As usual, we say that $\phi$ is an extension (from $F$ to $E$) of $\psi$ if $R(\phi)=\psi$, i.e., for all $x \in F$, we have $\phi(x)=\psi(x).$

The Hahn-Banach extension theorem tells us that we can always extend bounded linear functionals without increasing the norm. With $E$ and $F$ as above, we have:

Theorem (Hahn-Banach Extension Theorem for linear functionals on normed spaces) 

Let $\psi \in F^*$. Then there exists $\phi \in E^*$ with $\|\phi\|_\op=\|\psi\|_\op$ such that $\phi$ is an extension of $\psi$.

Comments

  • This is clearly the minimum possible norm for an extension of $\psi$ (since restriction cannot increase norms).
  • If $F$ is a dense linear subspace of $E$, the result follows immediately from a general result about extensions from $F$ to $\overline F$ of continuous linear maps from $F$ into a Banach space $Y$. In this case the extension is unique.
  • In general the extension $\phi$ need not be unique. 
  • The standard proof uses Zorn's Lemma to find a "maximal extension" in a sense that we will discuss later. However, if $E$ is separable, then we don't need Zorn's lemma.
  • The complex version (where $\F=\C$) can be deduced from the real version (where $\F=\R$). We'll discuss this shortly.
  • The key result we need is the case where $\F=\R$ and $F$ has codimension 1 in $E$. As long as we can always extend by one dimension without increasing the norm, Zorn's lemma will do the  rest. In the separable case, you can just use one inductive sequence of extensions, then extend to the union of this nested sequence of subspaces, and then extend to the closure of this union to finish.
  • In the next part we will look at the key issue of how to extend by one (real) dimension.

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