Fa.functional-analysis

This is the set of notes I wrote up for the talk I gave at the student analysis seminar on September 14.

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Recall that we adopt the notation of Agnew and Morse [AM38] and write $[\mathscr{L},\rho,f,X]$ 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 $\Lambda \in \mathscr{L}$ ;
$f$ is invariant under each $\Lambda$ , viz., $f(\Lambda x) = f(x)$ for each $\Lambda \in \mathscr{L}$ ;
$Y$ is invariant under each $\Lambda$ , viz., $\Lambda(Y) \subseteq Y$ for each $\Lambda \in \mathscr{L}$ .

Furthermore, if $[\mathscr{L},\rho,f,X]$ is satisfied, we write $\{\mathscr{L},\rho,f,X\}$ to denote the collection of extensions $F$ of $f$ on $X$ such that $F$ is still dominated by $\rho$ , and that

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

for all $\Lambda \in \mathscr{L}$ . In this notation, the classical Hahn-Banach theorem can be phrased as follows:

Theorem 1 (Hahn-Banach). Let $X$ be a real vector space, and $\mathscr{L}=\{I\}$ , where $I:X \to X$ is the identity mapping. If $[\mathscr{L},\rho,f,X]$ , then $\{\mathscr{L},\rho,f,X\}$ is nonempty.

Recall that a semigroup is a set with an associative binary operation. In the last talk, we have proved the following symmetry-invariant extension of Hahn-Banach:

Theorem 2 (Agnew-Morse). Let $\mathscr{L}$ be a semigroup of commuting endomorphisms on a real vector space $X$ . If $[\mathscr{L},\rho,f,X]$ , then $\{\mathscr{L},\rho,f,X]$ is nonempty.

For a proof, see pages 25-27 of Lax [Lax02].

We also recall the key example from the last talk. We write $X^*$ to denote the space of all real-valued sequences, and $X_b$ to denote the space $l^\infty(\mathbb{R})$ of all bounded real-valued sequences. The classical Banach limit extend the usual limit in the following manner:

Theorem 3. 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 $k \in \mathbb{N}$ ;
$\displaystyle \liminf_{n \to \infty} x_n \leq L( (x_n)_{n=1}^\infty ) \leq \limsup_{n \to \infty} x_n$ .

We now see that the above theorem is an easy consequence of the Agnew-Morse theorem. Indeed, the Agnew-Morse theorem furnishes an extension of the regular limit operator that is invariant under the translation operator. A slight modification of the Agnew-Morse theorem allows us to extend the limit operator even further. Continue reading »

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

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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.

Mark H. Kim

Interpolation on Lebesgue Spaces

The Hahn-Banach Theorem and Symmetry Invariance II

The Hahn-Banach Theorem and Symmetry Invariance I

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