Hahn-Banach Theorem/Complex Vector Space

Theorem

Let $X$ be a vector space over $\C$.

Let $p : X \to \R$ be a seminorm on $X$.

Let $X_0$ be a linear subspace of $X$.

Let $f_0 : X_0 \to \C$ be a linear functional such that:

$\cmod {\map {f_0} x} \le \map p x$ for each $x \in X_0$.

Then there exists a linear functional $f$ defined on the whole space $X$ which extends $f_0$ and satisfies:

$\cmod {\map f x} \le \map p x$ for each $x \in X$.

That is, there exists a linear functional $f : X \to \C$ such that:

$\cmod {\map f x} \le \map p x$ for each $x \in X$

and:

$\map f x = \map {f_0} x$ for each $x \in X_0$.

Corollary

Let $\struct {X, \norm \cdot}$ be a normed vector space over $\C$.

Let $X_0$ be a linear subspace of $X$.

Let $f_0 : X_0 \to \C$ be a bounded linear functional.

Then $f_0$ can be extended to a bounded linear functional $f : X \to \C$ with:

$\norm f_{X^\ast} = \norm {f_0}_{\paren {X_0}^\ast}$

where $\norm \cdot_{X^\ast}$ and $\norm \cdot_{\paren {X_0}^\ast}$ are the norms of the normed dual spaces $X^\ast$ and $\paren {X_0}^\ast$.

Proof

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Define $g_0 : X_0 \to \R$ by:

$\map {g_0} x = \map \Re {\map {f_0} x}$

for each $x \in X_0$.

By Real Part of Linear Functional is Linear Functional, $g_0$ is an $\R$-linear functional.

Also, define $h_0 : X_0 \to \R$ by:

$\map {h_0} x = \map \Im {\map {f_0} x}$

for each $x \in X$.

By Imaginary Part of Linear Functional is Linear Functional, $h_0$ is an $\R$-linear functional.

Now, for each $x \in X$, we have:

\(\ds \cmod {\map {g_0} x}\)

\(=\)

\(\ds \sqrt {\paren {\map {g_0} x}^2}\)

\(\ds \)

\(\le\)

\(\ds \sqrt {\paren {\map {g_0} x}^2 + \paren {\map {h_0} x}^2}\)

\(\ds \)

\(=\)

\(\ds \cmod {\map {f_0} x}\)

\(\ds \)

\(\le\)

\(\ds \map p x\)

Let $X_\R$ be the realification of $X$.

By Hahn-Banach Theorem: Real Vector Space on $X_\R$, there exists a $\R$-linear functional $g : X \to \R$ extending $g_0$ and satisfying:

$\cmod {\map g x} \le \map p x$

for each $x \in X$.

Define $f : X \to \C$ by:

$\map f x = \map g x - i \map g {i x}$

for each $x \in X$.

From Real Linear Functional is Real Part of Unique Linear Functional, we have that:

$f : X \to \C$ is a linear functional

and:

$\map g x = \map \Re {\map f x}$

for each $x \in X$.

We now want to show that $f$ extends $f_0$.

Let $F_0$ be the restriction of $f$ to $X_0$.

We want to show that:

$f_0 = F_0$

We have:

$\map \Re {\map f x} = \map g x$

for each $x \in X$.

Hence:

$\map \Re {\map f x} = \map {g_0} x = \map \Re {\map {f_0} x}$

for each $x \in X_0$.

So:

$\map \Re {\map {F_0} x} = \map \Re {\map {f_0} x}$

for each $x \in X_0$.

From Linear Functional on Complex Vector Space is Uniquely Determined by Real Part, it follows that:

$\map {F_0} x = \map {f_0} x$

for each $x \in X_0$.

So $f$ indeed extends $f_0$.

Now take $x \in X$.

Pick $\lambda \in \C$ such that:

$\cmod \lambda = 1$

and:

$\lambda \map f x = \cmod {\map f x}$

Then, since $f$ is $\C$-linear:

\(\ds \cmod {\map f x}\)

\(=\)

\(\ds \lambda \map f x\)

\(\ds \)

\(=\)

\(\ds \map f {\lambda x}\)

\(\ds \)

\(=\)

\(\ds \map g {\lambda x} - i \map g {i \lambda x}\)

Since:

$\cmod {\map f x} \in \R$

we have that:

$\map g {i \lambda x} = 0$

Finally, we have:

\(\ds \cmod {\map f x}\)

\(=\)

\(\ds \map g {\lambda x}\)

\(\ds \)

\(\le\)

\(\ds \map p {\lambda x}\)

\(\ds \)

\(=\)

\(\ds \cmod \lambda \map p x\)

Seminorm Axiom $\text N 2$: Positive Homogeneity

\(\ds \)

\(=\)

\(\ds \map p x\)

So $f$ is a linear functional satisfying our requirements.

$\blacksquare$

Source of Name

This entry was named for Hans Hahn and Stefan Banach.

Sources

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