The Hahn-Banach Theorem and Symmetry Invariance I

Sep 232011

This is the set of notes I wrote up for the talk I gave at the student analysis seminar on September 13.

* * * * *

Definition 1. Let $X$ be a real vector space. A sublinear functional on $X$ is a real-valued function $\rho$ on $X$ satisfying

Positive homogeneity. $\rho(a x) = a \rho(x)$ for all $a > 0$ and $x \in X$ ;
Subadditivity. $\rho(x+y) \leq \rho(x)+\rho(y)$ for all $x,y \in X$ .

A linear functional is a sublinear functional that is additive, viz.,

$\rho(x+y) = \rho(x)+\rho(y)$

for all $x,y \in X$ .

Theorem 2 (Hahn, 1927; Banach, 1932). If a linear functional $f$ on a subspace $Y$ of a real vector space $X$ is
dominated by a sublinear functional on $X$ , then there exists an extension of $f$ on $X$ still dominated by the same sublinear functional.

Proof. Extend the linear functional by one dimension and use transfinite induction.

We now discuss a well-known application of the Hahn-Banach theorem, known as Banach limits, which extends the notion of limit to a more general class of sequences.

Definition 3. $X_b = l^\infty(\mathbb{R})$ is the normed linear space of all bounded real-valued sequences with the norm

$\|(x_n)_{n=1}^\infty\| = \sup_{n \in \mathbb{N}} |x_n|$ .

Of course, not every bounded sequence converges. The classical Banach limit extends the notion of limit as follows:

Theorem 4. There exists an operator $L:X_b \to X_b$ , called a generalized limit or a Banach limit, such that

The generalized limit agrees with the usual limit if the usual limit exists;
$L ((x_n)_{n=1}^\infty + (y_n)_{n=1}^\infty) = L( (x_n)_{n=1}^\infty ) + L( (y_n)_{n=1}^\infty )$ ;
$L( (x_n)_{n=1}^\infty ) = L( (x_n)_{n=k}^\infty)$ for any $k \in \mathbb{N}$ ;
$\displaystyle \liminf_{n \to \infty} x_n \leq L( (x_n)_{n=1}^\infty ) \leq \limsup_{n \to \infty} x_n$ .

Proof. We define $p:X_b \to \mathbb{R}$ by setting

$\displaystyle p((x_n)_{n=1}^\infty) = \limsup_{n \to \infty} x_n$

This is a sublinear functional on $X_b$ . If $S$ is the left-translation map

$S((x_n)_{n=1}^\infty) = (x_n)_{n=2}^\infty$ ,

then $p(Sx)=p(x)$ for all $x \in X_b$ . Let $X_c$ be the space of convergent real-valued sequences, which is a linear subspace of $X_b$ . We define a linear functional $l:Y \to \mathbb{R}$ by

$\displaystyle l((y_n)_{n=1}^\infty) = \lim_{n \to \infty} y_n$ .

Then $l(y)=p(y)$ for all $y \in X_c$ , and $l(Sy)=l(y)$ for all $y \in X_c$ . We invoke the Hahn-Banach theorem to furnish an extension $L:X_b \to \mathbb{R}$ of $l$ such that $L$ is dominated by $p$ and that $L$ is invariant under left translation. In particular,

$\displaystyle \liminf_{n \to \infty} x_n \leq L((x_n)_{n=1}^\infty) \leq \limsup_{n \to \infty} x_n$ ,

for each $x \in X_b$ satisfies the inequality $-p(-x) \leq L(x) \leq p(x)$ . It thus follows that $L$ is the desired map. $\square$

Note that one of the key properties of the limit operator was “invariance under the translation operator,” viz.,

$L((x_n)_{n=k}^\infty) = L((x_n)_{n=1}^\infty)$ .

Keeping this in mind, we introduce a new notation, which will bring out the key elements of the Hahn-Banach theorem. The following is a slightly modified version of the notation employed in Agnew and Morse [AM38].

We shall write $[\mathfrak{L},\rho,f,X]$ to denote the following:

$X$ is a real vector space;
$\rho$ is a sublinear functional on $X$ ;
$f$ is a linear functional defined on a subspace $Y$ of $X$ ;
$f$ is dominated by $\rho$ , viz., $f(x) \leq \rho(x)$ for all $x \in Y$ ;
$\Lambda$ is a collection of endomorphisms on $X$ ;
$\rho$ is invariant under each $\Lambda$ , viz., $\rho(\Lambda x) =\rho(x)$ for each $\Lambda \in \mathfrak{L}$ .
$f$ is invariant under each $\Lambda$ , viz., $f(\Lambda x) = f(x)$ for each $\Lambda \in \mathfrak{L}$ ;
$Y$ is invariant under each $\Lambda$ , viz., $\Lambda(Y) \subseteq Y$ for each $\Lambda \in \mathfrak{L}$ .

If $[\mathfrak{L},\rho,f,X]$ , then we write $\{\mathfrak{L},\rho,f,X\}$ to denote the set of extensions $F$ of $f$ on $X$ such that $F$ is still dominated by $\rho$ , and that

$F(\Lambda x) = F(x)$

for all $\Lambda \in \mathfrak{L}$ . With the new shorthand, we can rephrase the classical Hahn-Banach theorem as follows:

Theorem 5 (Hahn-Banach, as stated in [AM38]). Let $X$ be a real vector space, and $\mathfrak{L}=\{I\}$ , where $I:X \to X$ is the identity mapping. If $[\mathfrak{L},\rho,f,X]$ , then $\{\mathfrak{L},\rho,f,X\}$ is nonempty.

The main theorem of the talk is the symmetry-invariant version of the Hahn-Banach theorem known as the Agnew-Morse theorem. The theorem was first proven by A. G. Agnew and A. P. Morse in 1938 [AM38], and was subsequently generalized by E. J. McShane, R. B. Warfield, and V. M. Warfield in 1969 [MWW69]. We state here the minor generalization of the McShane-Warfield-Warfield formulation of the theorem, as stated in [Lax02]:

Theorem 6 (Agnew-Morse, 1937; McShane-Warfield-Warfield, 1969; Lax, 2002). Let $X$ be a real vector space, and $\mathfrak{L}$ be a collection of endomorphisms on $X$ that commute, viz.,

$\Lambda_t \Lambda_s = \Lambda_s \Lambda_t$

for any pair of operators $\Lambda_t$ and $\Lambda_s$ in $\mathfrak{L}$ . If $[\mathfrak{L},\rho,f,X]$ , then $\{\mathfrak{L},\rho,f,X\}$ is nonempty.

A remark is in order. If $f$ is invariant under a pair of operators $\Lambda_t, \Lambda_s \in \mathfrak{L}$ , then $f$ is also invariant under $\Lambda_t \Lambda_s$ . Moreover, if $\Lambda_t$ and $\Lambda_s$ commute with every operator in $\mathfrak{L}$ , then so does $\Lambda_t\Lambda_s$ . We may thus enlarge $\mathfrak{L}$ by adding the identity $I$ and all finite products of the elements therein; this turns $\mathfrak{L}$ into a semigroup, which we now define:

Definition 7. A semigroup is a set $S$ with an associative binary operation.

For example, $[0,\infty)$ with the usual addition is a semigroup. We note that the term “semigroup” in functional analysis usually refers to the following:

Definition 8. A one-parameter semigroup of operators over a Banach space $latexX$ is a family $\{X \xrightarrow{L_t} X\}_{t \geq 0}$ of bounded operators such that

$L(0) = I$ ;
$L_{t+s} = L_tL_s$ for all $t,s \geq 0$.

Note that a one-parameter semigroup is essentially a “representation” of the semigroup $[0,\infty)$ on the Banach space $X$ : it is a homomorphism of $[0,\infty)$ into the collection $\mathrm{End}(X)$ of endomorphisms on $X$ .

Let us now return to the main theorem. We have shown that proving Theorem 6 amounts to proving the following variant:

Theorem 9 (Agnew-Morse, the semigroup version)
Let $\mathfrak{L}$ be a semigroup of commuting endomorphisms on a real vector space $X$ . If $[\mathfrak{L},\rho,f,X]$ , then $\{\mathfrak{L},\rho,f,X\}$ is nonempty.

This is the statement of the Agnew-Morse found in [MWW69]. The following proof is taken from [Lax02], pp.25-27:

Proof. Let $\Gamma$ denote a convex combination of operators in $\mathfrak{L}$ , viz.,

$\Gamma = \sum_n a_n \Lambda_n$ ,

where $(a_n)_{n=1}^\infty$ is a nonnegative sequence whose sum is 1, and $(\Lambda_n)_{n=1}^\infty$ a sequence of operators in $\mathfrak{L}$ . We set

$\sigma(x) = \inf \rho(\Gamma x)$ ,

where the infimum is taken over all convex combinations $\Gamma$ . This is our bounding sublinear functional, as we shall show.

Note that the convex hull of $\mathfrak{L}$ , the collection of all convex combinations $\Gamma$ , closed under composition. For each convex combination $\Gamma$ , we have the following inequality:

$\rho(\Gamma x) = \rho \left( \sum_n a_n \Lambda_n x \right) \leq \sum_n a_n \rho \left( \Lambda_n x \right) = \sum_n a_n \rho(x) = \rho(x)$ . (Eq. 1)

Therefore, $\sigma(x) \leq \rho(x)$ .

We claim that $\sigma$ is a sublinear functional. Indeed,

$\sigma(a x)= \inf \rho(\Gamma a x)= \inf \rho(a \Gamma x)= a \inf \rho(\Gamma x)= a \sigma(x)$ ,

and so $\sigma$ is positive homogeneous. To see that $\sigma$ is subadditive, we pick arbitrary elements $x$ and $y$ of $X$ . By the definition of $\sigma$ , each $\varepsilon>0$ furnishes a pair of maps $\Gamma_1$ and $\Gamma_2$ in the convex hull of $\mathfrak{L}$ such that

$\rho(\Gamma_1 x) \leq \sigma(x) + \varepsilon$ and $\rho(\Gamma_2 x) \leq \sigma(x) + \varepsilon$ . (Eq. 2)

Since $\Gamma_1$ and $\Gamma_2$ commute, we have

$\sigma(x+y) \leq \rho(\Gamma_1\Gamma_2(x+y)) = \rho(\Gamma_2\Gamma_1 x + \Gamma_1 \Gamma_2 y)$ .

The subadditivity of $\rho$ implies that

$\rho(\Gamma_2\Gamma_1 x + \Gamma_1 \Gamma_2 y) \leq \rho(\Gamma_2\Gamma_1 x) + \rho(\Gamma_1\Gamma_2 y)$ ,

and

$\rho(\Gamma_2\Gamma_1 x) + \rho(\Gamma_1\Gamma_2 y) \leq \rho(\Gamma_1 x) + \rho(\Gamma_2 y)$

by (Eq. 1). It thus follows from the estimate (Eq. 2) that

$\sigma(x+y) \leq \sigma(x) + \sigma(y) + 2\varepsilon$ ,

whence $\sigma$ is subadditive as was claimed.

We now show that $\sigma$ dominates $f$ . By the invariance of $f$ and $Y$ under each operator in $\mathfrak{L}$ , we have the following equality for each convex combination $\Gamma$ and each element $y$ of $laetx Y$:

$f(\Gamma y) = f \left( \sum_n a_n \Lambda_n y \right) = \sum_n a_n f(\Lambda_n y) = \sum_n a_n f(y) = f(y)$ ;

that is, $f$ is invariant under each $\Gamma$ . Since $Y$ is invariant under each $\Gamma$ as well, we have

$\rho(\Gamma y) \geq f(\Gamma y) = f(y)$

for each $y \in Y$ . It thus follows that

$f(y) \leq \inf \rho(\Gamma y) = \sigma(y)$

for all $y \in Y$ , whence by the Hahn-Banach theorem that there is an extension $F$ of $f$ on $X$ still dominated by $\sigma$ .

We claim that $F$ is invariant under all operators $\Lambda$ in $\mathfrak{L}$ . To show this, we use an averaging trick. For a fixed $\Lambda \in \mathfrak{L}$ , we set

$\displaystyle \Gamma_N = \frac{1}{N} \sum_{n=0}^{N-1} \Lambda^n$

for each $N \in \mathbb{N}$ . Since $\mathfrak{L}$ is a semigroup, $\Gamma_N$ is in the convex hull of $\mathfrak{L}$ . Therefore, we have

$\displaystyle \sigma(x-\Lambda x) \leq \rho(\Gamma_N(x-\Lambda x)) =\rho(\Gamma_N(I-A)x)=\frac{1}{N} \rho(x - \Lambda^N x)$

for each $x \in X$ . The last equality follows from

$\displaystyle \Gamma_N(I-\Gamma) = \frac{1}{N} (I-\Gamma^N)$ ,

which is just the geometric series. The subadditivity of $\rho$ implies that

$\displaystyle \frac{1}{N} \rho(x - \Lambda^N x) \leq \frac{1}{N} \left[ \rho(x) +\rho(-\Lambda^n x) \right]$ ,

and we have

$\frac{1}{N} \left[ \rho(x) +\rho(-\Lambda^n x) \right] \leq \frac{1}{N} \left[ \rho(x) + \rho(-x) \right]$

by the invariance of $\rho$ under all operators in $\mathfrak{L}$ . It thus follows that

$\displaystyle \sigma(x-\Lambda x) \leq \frac{1}{N} \left[ \rho(x) + \rho(-x) \right]$ .

Letting $N \to \infty$ , we now see that

$\sigma(x-\Lambda x) \leq 0$ .

Since $\sigma$ dominates $F$ , the above inequality implies that

$F(x-\Lambda x) \leq 0$ ,

whence by linearity

$F(x) \leq F(\Lambda x)$ .

Replacing $x$ with $-x$ , we also have

$F(-x) \leq F(-\Lambda x)$ ,

and so

$0 \leq F(x - \Lambda x)$ .

It thus follows that $F(x- \Lambda x) = 0$ , or

$F(x) = F(\Lambda x)$ ,

as was claimed.

It now suffices to observe that $\rho$ dominates $\sigma$ , hence $F$ . $\square$

Next time, we shall consider a few applications of the Agnew-Morse theorem.

References

[AM38] A. G. Agnew and A. P. Morse, “Extensions of linear functionals, with applications to limits, integrals, measures, and densities,” Annals of Mathematics 39 (1938), no. 1, 20-30.
[Lax02] Peter D. Lax, Functional Analysis, John Wiley & Sons. 2002.
[MWW69] E. J. McShane, R. B. Warfield, and V. M. Warfield. “Invariant extensions of linear functionals, with applications to measures and stochastic processes,” Pacific Journal of Mathematics 28 (1969), no. 1, 121-142.

Mark H. Kim

The Hahn-Banach Theorem and Symmetry Invariance I

Leave a Reply Cancel reply

Recent Blog Posts

Recent Comments

Categories

Blogroll

Internets, The

Mathematics

Mark H. Kim

The Hahn-Banach Theorem and Symmetry Invariance I

Leave a Reply Cancel reply

Recent Blog Posts

Recent Comments

Categories

Tags

Blogroll

Internets, The

Mathematics