5.1. Counterparts of Jacobi and Parallelogram Identities
Approximately a century before the appearance of the Jordan-von-Neumann parallelogram criterion, K. G. Jacobi established a remarkable identity relating the moment of inertia of a system of mass (with respect to its barycentre) and the distances between them. Namely, in a Hilbert space
H, for every
and
with
and
, one has
In Part
of the next remark, we shall see that the Jacobi identity, implying the parallelogram identity, characterizes the inner product spaces as well. Therefore, considering its counterparts, we must deal with inequalities. Having in mind the future use, we define the Jacobi and adjoint classes of normed spaces.
Definition 5.1. Let X be a Banach space, and . We say that the space X belongs to the Jacobi class , or the adjoint Jacobi class , if, for every X-valued stochastic variables ξ and η that are independent and identically distributed on a probability measure space , one has, respectively, the estimates Let also and . For a purely atomic probability space , let We assume that for the measures μ with continuous components. It is convenient to use a simplex representation for the set of purely atomic probability measures. For a finite or countable set I, assume that is the set of all satisfying .
We say that the space X belongs to the atomic Jacobi class , or the adjoint atomic Jacobi class , if, for every finite or countable I and , one has, respectively,
Let also and . For and , we designate the best constants and by and correspondingly. As we shall see in Theorem , the purely atomic and continuous classes coincide, leaving us with the notations and .
Let be a metric space, and . We say that the space E belongs to the Jacobi class , if, for every and E-valued stochastic variables ξ and η that are independent and identically distributed on a probability measure space , one has the estimate c We shall use the constants
and
to estimate the self-Jung constant (see Theorem
below and related references) and the Kottman [
50] constant
(see Definition
;
is the unit ball of X; compare with [
11]), to identify the existence of isometric extensions of the Hölder mappings between various spaces in
Section 6 and estimate the Dol’nikov-Pichugov and mutual diameter constants in
Section 5.2.
Remark 5.1.
One clearly has the inclusions .
Let us note that, if a Banach space X is in some Jacobi, or adjoint Jacobi class, and Y is either a subspace or a quotient, or finitely represented (see Definition in X, then Y is in the same class.
Restricting ourselves by only finite index sets I in the definition of the atomic Jacobi classes, we obtain an equivalent definition of the same classes. The same assertion holds for the general Jacobi classes thanks to Part of Theorem .
According to Parts and of the next theorem, all the nontrivial Jacobi classes contain superreflexive spaces only.
A Banach space X is a Hilbert space if, and only if, (i.e., holds). Indeed, choosing and for arbitrary , we obtain the parallelogram identity If , the Hölder inequality provides the inclusions The triangle inequality implies that every metric space belongs to the class for .
For
and a probability space
, we shall deal with the Bochner spaces
and
, where the latter space is just a convenient representation for independent identically distributed stochastic variables
ξ and
η in the form
for an arbitrary
that is also convenient to apply the complex interpolation and duality theories of linear operators.
Let
and
be the difference operator and its adjoint, respectively, defined by
Lemma 5.1. Let . Then one has , , , and is a bounded projector with . If either μ is purely atomic, or X and possess the Radon-Nikodým property and , then the operator is the dual of .
Proof of Lemma 5.1. The Radon-Nikodým property requirement provides the relations between the corresponding Lebesgue-Bochner spaces in the continuous setting. The Minkowski and Jensen inequalities imply and, hence, the estimate for the norm of P. The validity of and is checked straightforward and implies . Similarly, with the aid of the identities and , we see that delivering both the second identity of the lemma and . ☐
Corollary 5.1. Let be a probability measure space with either μ being purely atomic uniform measure on the finite discrete Ω, or μ being uniform (continuous) on Ω, and let be a compatible pair of Banach spaces. Assume also that is a positive weight on with respect to on with Let also , , and . Then we have where and the intersections inherit the norms. Proof of Corollary 5.1. Part
of Theorem
provides the isometry
Thus, to finish the proof it is sufficient to check the validity of the conditions of Theorem
on the interpolation of subspaces. According to Lemma
,
(
i.e., defined for the uniform distribution on
) is a projector onto
with
The conditions on the weights imply that the identity operator is the isomorphism
Now
and
provide the desirable estimate
justifying the applicability of Theorem
. ☐
Regarding Parts and of the next theorem, let us note that there are reflexive (and uniformly convex in every direction) spaces with trivial Jacobi constants. One example is the -sum () of () with either or .
Theorem 5.1. Let X be a Banach space, let E be a metric space, and . Then one has:
and if ;
if , and if ;
if , and if ;
if , then , and the same holds for the general J and ;
if either with or with , then X is superreflexive; and, if , then and ;
for ;
for ;
and ;
and if, for an increasing , X contains either or almost isometrically for every k.
Moreover, Part shows the coincidence of the atomic and non-atomic classes: and .
Proof of Theorem 5.1. Part
for
follows either from the boundedness of the operators
D and the restriction of
onto
established in the proof of Lemma
with the aid of the triangle, Jensen and Minkowski inequalities (one can also use the Hölder inequality
to avoid the restriction step), or with the aid of the complex interpolation method (in the case of X) applied to the pairs
where we also use Corollary
and Theorem
. Part
of Remark
(or Hölder inequality) finishes the proof of
.
The duality (Part ) for the case of purely atomic measures follows immediately from Lemma : is the dual of D and and the fact that X is a subspace of .
In the case of Part with the aid of this duality, we have one of the inclusions or implying, thanks to the Theorem , the estimates for the self-Jung constants (the uniform normal structure for either or X) or , which, in turn, are followed by the superreflexivity of X and provided by Part of Theorem . At the same time, Part of Theorem states that if X is not reflexive. This observation provides the second half of by reversing the implications.
Since the reflexivity implies the Radon-Nikodým property (see
Section 7), the rest of Part
follows either from Part
in the trivial case
, or from the duality in Lemma
.
To establish Parts
and
in the case of Banach spaces, let us assume that
is a countable set of atoms with
. We also take
and
for
and obtain
Tending
, we see that, necessarily,
in Parts
and
. Moreover, when
, we also obtain
and
. Since
is a Hilbert space, we have the Jacobi identities
Interpolating these inequalities with the limiting cases
and
provided by and as in Part
by means of the complex method, we obtain the estimates
and
. Considering
with an even
n with the uniform probability
and
, we see that
providing the identities
and
. To finish the proof of
and
for Banach spaces it is left to notice that every Banach space contains an isometric copy of
. To establish Part
for metric spaces, we similarly take
to be composed of two atoms with
. One also considers
for
and
to establish a counterpart of
:
implying the rest of
.
Parts and are established by means of the complex interpolation, correspondingly, between the inclusions and and the inclusions in Part (valid for every X). As in Part , we use Theorem and Corollary .
The nontrivial inclusions and in Part h are established by approximating a general stochastic variable ξ with simple functions (stochastic variables).
If X contains
almost isometrically, then, according to Part
of Remark
,
Choosing the discrete
with
and
from Lemma
and, then,
with
and
from Lemma
(the Hadamard function
h is found in Definition
), we obtain
implying the first estimate in Part
. To finish the proof of the theorem, we obtain in the same manner
and
☐
Corollary 5.2. The proof of Theorem 5.1 also shows that for , we have There is a huge amount of literature dedicated to the Clarkson inequalities and most of the rest of this subsection (if not all) should be known in, most probably, different terms. The proof and applications of the classical Clarkson inequalities can be found, for example, in [
1]. We intend to achieve only the level of completeness suitable for our purposes.
Definition 5.2. Let X be a Banach space and , . We say that X is in Clarkson class if for every ; For , we designate the best constant by .
As shown in Part
of Theorem
and Part
of Theorem
in [
2], the Clarkson constant
dominates some other important constants, including the non-squareness constant of James.
Lemma 5.2. Let X be a Banach space and , . Then we have
;
;
;
;
;
for .
Proof of Lemma 5.2. Part
follows from the triangle and Hölder inequalities
Part
is the outcome of the complex interpolation of the operator
whose boundedness at the end-point spaces of the corresponding interpolation pairs is interpreted as
. In the same manner, the upper estimate in
is established with the aid of the complex interpolation of the same operator between the inclusions
,
and
.
The first inequality in Part
follows from the isometric inclusion
. The lower estimate for
is found by choosing
in the inequality
The combination of this lower estimate with the upper estimate established for Part
finishes the proof of both
and
.
Let us treat Part
. Since X is a subspace of
, it is enough to establish the implication
Observing the isometry
for
and a finite
I, we choose
from the unit sphere of
satisfying, for a given
and
, the estimates
To establish it is enough to note that . ☐
Note that the Minkowski inequality implies that the class contains all Banach spaces too.
Remark 5.2. Parts of Theorem 5.1 and Lemma 5.2 (and the majority of the results below) explain the importance of the sharpness of the Jacobi and Clarkson constants. In this article we apply the approach leading to precise constants for limited ranges of parameters. An alternative and somewhat less precise approach based on our counterparts of the Pythagorean theorem for Birkhoff-Fortet orthogonality is presented in [2]. 5.1.1. Constructive Approach for Particular Spaces
In this subsection, we establish sharp results for -spaces, their subspaces, quotients and the spaces finitely represented (see Definition in them.
The next simple lemma follows from the Minkowski and triangle inequalities. It provides slight improvement with respect to the inclusion in the case of the purely atomic measure μ with a finite number of atoms. Recall that is in Definition .
Lemma 5.3. For a Banach space X and a probability measure space , one has
.
The next theorem (due to Riesz and Thorin [
45]) is well-known and is also an immediate corollary of the easier case of Banach spaces of both parts of Theorem
.
Theorem 5.2 ([
45]).
For , and , one has In the case
with
, the inequalities in Parts
,
of the next theorem are due to Wells, Williams and Hayden (see [
9,
10,
11]).
Theorem 5.3. Let Y be a Banach space from the class with and finite or countable set . Let also X be either finitely represented (see Definition 7.1) in Y or a subspace, or a quotient of Y. Assume also that , with In addition, let . Then the following inequalities take place: for , and for , and for , and for , and Moreover, we also have for :
One has equalities in and if, for an increasing , X contains either or almost isometrically for every k.
Remark 5.3
In an aggregated form (if the last assumption holds as well) the statement of Theorem 5.3 can be written as for . It particular, for , and an infinite-dimensional , one has According to the idea of Section 3, this theorem, combined with Theorem and the corresponding Rademacher type-cotype descriptions (see Theorems and in [2]), provides, in particular, the exact values of the Jacobi constants for various classes of anisotropic Besov, Lizorkin-Triebel and Sobolev spaces and their duals and the -sums of function spaces, non-commutative spaces and their duals. Proof of Theorem 5.3. Thanks to Lemma 2.3 combined with Theorem , we shall deal mostly with the spaces. These spaces will possess the same tree type as the space X itself but different values of the parameters of the spaces at the vertexes of T. As in Definition , the parameter functions are defined on one and the same parameter position set determined by T and the spaces at its vertexes. Thus, . The space of the same tree type T with the parameter function p will be designated by .
Assume that is the parameter function of Y as an -space. Let also be a constant parameter function equal to 2 with the same domain as p (shared also by all the other parameter functions). Noting that the subspaces, quotients and the spaces finitely represented in Y inherit those properties of Y that are mentioned in Parts , we may assume that , while Lemma further allows to assume .
Since the extreme cases
and
are covered by more general Lemma
, we assume that
and
. Next we observe that one can find a parameter function
with the values in
, satisfying
for some
, exactly when either
The last preliminary observation is that, in a Hilbert space
, one has
meaning
. Here we assume that
I and
are endowed with the probability measures
for
and
respectively. Thus, (the Banach case of) Theorem
and Corollary
provide the identities
where
is chosen as in
and depends on
r. The properties of the complex interpolation method for
D provide the upper estimates in
and
. In the case of the upper estimates in
and
, we also involve Theorem
to interpolate the images of the projector
P (
i.e.,
D) from Lemma
and, then, apply the interpolation estimates to the restriction of
.
The lower estimates that are parts of the equalities for
and
in
are provided by the isometric inclusion
and Parts
and
of Theorem
. Together with
, they imply the identities in
. The inequalities in
and
follow from the complex interpolation of the pairs
with
:
The potential equalities in and are provided by Part of Theorem . ☐
Theorem 5.4. Let with , where is a von Neumann algebra with a normal semifinite faithful weight τ, and let be finite or countable set. Let also X be either finitely represented (see Definition in Y or a subspace, or a quotient of Y. Assume also that , with Then the following inequalities take place: for , and for , and for , and for , and Moreover, we also have for : One has equalities in and if is infinite-dimensional. Proof of Theorem 5.4. This proof is a simplification of the proof of Theorem corresponding to repeating its proof for the particular case of a Lebesgue space. The first difference is that we use Part of Theorem instead of the complex interpolation of -spaces (Corollary ) to cover the case of semifinite algebras and, then, use Theorem to cover the general case with the same constants. The second difference is the usage of Lemma instead of Part of Theorem to establish the potential equalities in and when and . ☐
The next theorem is the counterpart of the previous one corresponding to the Clarkson classes.
Theorem 5.5. Let Y be a Banach space from the class with . Let also X be either finitely represented (see Definition in Y or a subspace, or a quotient of Y. Then one has:
with the sharp constant for for and the constant is sharp if . Remark 5.4.
In an aggregated form, Theorem states that, for either we have In particular, one has for . According to the idea of Section 3, this theorem, combined with Lemma and the corresponding Rademacher type-cotype descriptions (see Theorems and in [2]), provides, in particular, the exact values of the Clarkson constants for various classes of anisotropic Besov, Lizorkin-Triebel and Sobolev spaces, their duals and the -sums of function spaces, non-commutative spaces and their duals. Proof of Theorem 5.5. All the lower estimates related to the sharpness in Part
are provided by Part
of Lemma
, while the upper estimates hold thanks to the same complex interpolation procedure as is employed in the proof of Theorem
applied to the operator
T from the proof of Lemma
. To establish the lower estimates (the sharpness) in Part
, we note that Y contains an isometric copy of
for every
, including either
or
with
and
or
with
. Considering the pairs
and
of elements
in both
and
, we see that either
implying
under the conditions of Part
. ☐
Theorem 5.6. Let with , where is a von Neumann algebra with a normal semifinite faithful weight τ. Let also X be either finitely represented (see Definition in Y or a subspace, or a quotient of Y. Then one has:
with the sharp constant for for and the constant is sharp if . Proof of Theorem 5.6. In the same way as the proof of Theorem relates to the one of Theorem , this proof is a simplification of the proof of Theorem corresponding to repeating its proof for the particular case of a Lebesgue space. The only difference is that we use Part of Theorem instead of the complex interpolation of -spaces to cover the case of semifinite algebras and, then, use Theorem to cover the general case with the same constants. ☐
5.2. Quantitative Hahn-Banach Theorems
In this section we provide lower estimates for the distance between two bounded (non-convex) subsets of a Banach space sufficient for the existence of a separating hyperplane. This problem was considered by Dol’nikov [
3] and Pichugov [
4] in the settings of the Hilbert and Lebesgue spaces correspondingly. Pichugov’s approach was relying on new inequalities for the Lebesgue spaces. In the first subsection we extend Pichugov’s inequalities to a wide range of function, independently generated and noncommutative
-spaces and establish adjoint counterparts of these inequalities.
5.2.1. Pichugov Classes
In this subsection we introduce Pichugov and adjoint Pichugov classes of Banach spaces, investigate their properties, evaluate and calculate the corresponding constants and study their relations with the Jacobi and Clarkson classes of Banach spaces introduced earlier.
Definition 5.3. Let X be a Banach space, and . We say that the space X belongs to the adjoint Pichugov class , or the Pichugov class , if, for every combination ξ, η, and of independent X-valued stochastic variables, such that the variables in the pairs ξ and and η and are (independent and) identically distributed on a probability measure space , one has, respectively, the estimates For and , we designate the best constants and by and correspondingly.
We shall use the constants and to estimate the Dol’nikov-Pichugov and the mutual diameter constants in Theorems – to provide the quantitative description for the Hahn-Banach separation.
Remark 5.5.
As with Jacobi classes, restricting ourselves by only finite index sets I in the definition of the Pichugov classes, we obtain an equivalent definition of the same classes. The motivation (proof) is the same as in Part of Remark .
Thanks to , let us note that, if a Banach space X is in some Pichugov, or adjoint Pichugov class, and Y is either a subspace or a quotient, or finitely represented (see Definition in X, then Y is in the same class.
According to Part of the next theorem some Pichugov classes contain superreflexive spaces only.
Letting and in the definition of the Pichugov classes, we obtain the inclusions A Banach space X is a Hilbert space if, and only if, . Indeed, this statement is reduced to Part of Remark thanks to the inclusions in .
Theorem 5.7. Let X be a Banach space, and . Then one has:
and ;
;
if either with or with , then X is superreflexive;
and if, for an increasing , X contains either or almost isometrically for every k.
Proof of Theorem 5.7. Part
follows from the inequalities
where we used the observation that
and
have identical distributions (independent of
ξ and
) and applied the definition of Jacobi classes to the stochastic variable
.
With the aid of Parts of Theorem and Lemma , Part implies . Thanks to Part of Remark , we derive Part from Part of Theorem . Part follows from Parts and of Theorem with the aid of Part of Remark .
The cases
and
for
of Parts
and
of the next theorem were established by Pichugov [
4] but he used the scheme of Wells and Williams [
11] to establish the upper estimates of the constants and completely different approach to their sharpness.
Theorem 5.8. Let Y be a Banach space from the class with . Let also X be either finitely represented (see Definition in Y or a subspace, or a quotient of Y. Then we have:
;
for , ;
for , .
Proof of Theorem 5.8. Part follows from the inclusions in Part of Theorem and Remarks and . In turn, Part implies the upper estimates in and , while the lower estimates are provided by the identities and (under the conditions in and correspondingly) and the sharpness of the corresponding Jacobi constants established in Parts and of Theorem . ☐
In exactly the same manner (even with simplifications), we deduce the following theorem from Theorems and .
Theorem 5.9. Let with , where is a von Neumann algebra with a normal semifinite faithful weight τ. Let also X be either finitely represented (see Definition in Y or a subspace, or a quotient of Y. Then one has:
;
for , ;
for , .
5.2.2. Dol’nikov-Pichugov and Mutual Diameter Constants
The Dol’nikov-Pichugov constant for a particular Banach space X quantifies the corresponding Hahn-Banach separability theorem for X governing the separability of bounded (non-convex) sets by a hyperplane and, even, quantifying the corresponding strict separability.
Definition 5.4. Let X be a Banach space.
For a bounded subset , let its diameter be Given, in addition, a subset , the Chebyshev radius of A relative to B is is the Chebyshev radius of A. For an infinite subset , let its separation be For a bounded subset , its measure of noncompactness (see [51,52,53] and Section 11.1.2 in [2]) is or bounded subsets , let its mutual diameter be For subsets , the distance between them is For , we define the Dol’nikov-Pichugov constant by means of the relation Let also the mutual diameter constant be The fact that
is indeed a measure of noncomactness is Erzakova’s celebrated result (see [
51,
52,
53]).
In fact, the Dol’nikov-Pichugov constant can be computed relying on finite subsets only and, thus, is a local parameter as shown in the next lemma.
Lemma 5.4. Let X and Y be Banach spaces and .
;
;
, and if X is finitely represented (see Definistion in Y;
if , then X is superreflexive;
if , then .
Proof of Lemma 5.4. Part
follows from the observation that, if
, then
for some finite
and
. Indeed, we have
Now Part
implies
because
. The definition provides the first half of
, implying also the second half with the aid of
. We deduce
from Part
of Theorem
with the aid of the estimate
provided by Part
of Theorem
. In the same manner, the combination of the same inequality and Part
of Theorem
implies
, finishing the proof. ☐
To deal with the Dol’nikov-Pichugov constant, we shall need the Kirchberger theorem [
54,
55] in an equivalent form convenient for our purposes.
Theorem 5.10 (Kirchberger). For , let X be an n-dimensional normed space, and be its finite subsets. Then if, and only if, for every with .
To formulate the following theorem devoted to the upper estimate for the Dol’nikov-Pichugov constant, for
and
, we define the function
by means of
The next theorem (without the upper estimate for the mutual diameter in Part
that we have just introduced) is a generalization of the related results of Dol’nikov [
56] and Pichugov [
4], where the case
and
has been considered. Let us recall that every Banach space is contained in some Pichugov and adjoint Pichugov classes according to Part
of Theorem
.
Theorem 5.11. Let X be a Banach space and . Assume also that A and B are bounded subsets of X with and . Then we have:
, and, if and ,
;
;
if , .
Remark 5.6. Pichugov [4] has also proved that, for and an arbitrary Banach space X, one has In the case , he also provided the examples of bounded with and Proof of Theorem 5.11. In the case of
, let us assume that
,
i.e., there exist finite
and
and weights
with
and
. Hence, we have, according to the definition of the Pichugov class
,
Thus, the
-half of
follows.
To finish the proof of
, let us assume that
,
and
with
and
. For arbitrary weights
and
, we have, thanks to the definition of
,
where
z is chosen for the last inequality to hold that is possible thanks to the definition of the mutual diameter
. Since
is arbitrary, we establish the rest of
.
To approach Part
, following the related idea from [
4], we choose
,
and
, such that
and consider the translate
of
B. Since
, the sets
and
are not empty and
and
because the latter sets are extreme and contain
.
As in
, we select finite subsets
and
with
. Now the Kirchberger Theorem
applied to the sets
and
provides us with their subsets
and
with
where one uses the relation
.
Repeating the considerations from the proof of Part
leading to
with
and
instead of
A and
B and
, we obtain
Now the proof of
is finished with the aid of the triangle inequality.
Let us now reduce Part
to
. For an arbitrary
, we find
and
and finite
and
satisfying
,
,
Applying now Part
to the sets
and
in their linear span, we obtain that
Tending
ε to 0, one finishes the proof of the theorem. ☐
Combining Theorems and , we obtain particular cases (optimal choice of r) of the quantitative Hahn-Banach theorem for -spaces, their subspaces, quotients and Banach spaces finitely represented in them. In particular, it covers various classes of anisotropic Besov, Lizorkin-Triebel and Sobolev spaces and their duals and the -sums of function spaces and their duals.
Theorem 5.12 Let Y be a Banach space from the class with . Let also X be either finitely represented (see Definition in Y or a subspace, or a quotient of Y. Assume also that . Then we have:
and, if and ,
;
;
if , .