Abstract
Full Text
UDC 513.88:513.83
MATHEMATICS
V. D. MILMAN
ON A TRANSFORMATION OF CONVEX FUNCTIONS AND THE DUALITY OF THE \(\beta\)- AND \(\delta\)-CHARACTERISTICS OF A \(B\)-SPACE
(Presented by Academician L. V. Kantorovich on 12 XI 1968)
1. Let \(\mathfrak{B}\) be some family of subspaces of the Banach space \(B\). Consider the following averages of real bounded functions \(f(x)\) for \(x \in S(B)=\{x\in B:\|x\|=1\}\):
\[ \beta [f;\mathfrak{B}]=\sup_{E\in\mathfrak{B}}\inf_{x\in S(E)} f(x);\qquad \delta [f;\mathfrak{B}]=\inf_{E\in\mathfrak{B}}\sup_{x\in S(E)} f(x). \tag{1} \]
In the case when \(\mathfrak{B}\) is the family of all subspaces of finite defect, we shall write \(\beta[f,B]=\beta f\) and \(\delta[f,B]=\delta f\). In papers \((^{1,2})\) the averages (1) were studied for the special function \(f(\varepsilon;x,y)=\|x+\varepsilon y\|-1\) with respect to the variable \(y\in S(B)\) (here \(\beta_y[f,\mathfrak{B}]\) is denoted by \(\beta^0(\varepsilon;x,\mathfrak{B})\) and \(\delta_y[f,\mathfrak{B}]=\delta^0(\varepsilon;x,\mathfrak{B})\)) and the subsequent averaging of the obtained functions \(\beta^0\) and \(\delta^0\) with respect to the variable \(x\in S(B)\):
\(\beta\beta(\varepsilon;B)=\beta[\beta^0(\varepsilon;x,B),B]\), \(\delta\delta(\varepsilon;B)=\delta[\delta^0(\varepsilon;x,B),B]\).
In the present paper (Sec. 4) the relation between the \(\beta\)- and \(\delta\)-characteristics of a space and of its conjugate is studied. In this connection there arises a functional transformation (see Sec. 2) close to the Legendre transform:
\(\psi(\eta)=\sup\{\xi\eta-\varphi(\xi):\xi\ge 0\}=L_0(\varphi)\). Preliminarily, in Sec. 3 special minimal systems are considered, and in Sec. 5—some results on complementability.
In what follows, \(E(M)\) denotes the closed linear span of \(M\); \(B_1\simeq B_2\) means an isomorphism of the spaces \(B_1\) and \(B_2\); \(\{x_k\}_1^\infty\simeq\{y_k\}_1^\infty\) means that the sequences \(X=\{x_k\}_1^\infty\) and \(Y=\{y_k\}_1^\infty\) are isomorphic, i.e. the bounded linear operators \(T:E(X)\to E(Y)\) and \(T^{-1}\) exist, where \(\{Tx_k=y_k\}_1^\infty\). A sequence \(\{x_k\}_1^\infty\subset B\) is called minimal if there exists a conjugate system of functionals \(\{f_k\}_1^\infty\subset B^*\), \(f_k(x_j)=\delta_{kj}\). A basis \(\{x_k\}_1^\infty\subset B\) is called orthogonal if for any \(m\) and \(n\) \((m>n)\)
\[ \left\|\sum_n^\infty a_k x_k\right\|\ge \left\|\sum_m^\infty a_k x_k\right\|. \]
2. On a transformation of convex functions.
Theorem 1. The transformation
\[ \psi(\xi)=\mathcal{L}_1[\varphi(\eta)]=\sup_{\eta\ge0}\frac{\xi\eta-\varphi(\eta)}{1+\varphi(\eta)}, \tag{2} \]
where \(1+\varphi(\eta)>0\) for \(\eta\ge0\), has the following properties:
a) for \(\xi\ge0\), \(\psi(\xi)\) is a convex nondecreasing function, \(1+\psi(\xi)\ge0\), and the function \((\psi(\xi)+1)/\xi\) does not increase;
b) inversion formula: in the class of functions satisfying property a), there exists a unique function \(\varphi(\eta)=\mathcal{L}_1[\psi(\xi)]\) for which (2) is fulfilled;
c) for an arbitrary function \(\varphi(\eta)\), the inverse transform \(\mathcal{L}_1[\psi]=\mathcal{L}_1[\mathcal{L}_1(\varphi)]\) gives the function
\[
\widehat{\varphi}(\eta)=\sup\{r(\eta): r(\eta)\le \varphi(\eta)\ \text{and } r(\eta)\text{ satisfies property a)}\}.
\]
Examples. For \(p\ge1\),
\[
\mathcal{L}_1[(1+\eta^p)^{1/p}-1]=(1+\xi^q)^{1/q}-1,
\]
where
\[
p^{-1}+q^{-1}=1.
\]
3. Normalizing families of subspaces. A family \(\mathfrak{B}=\{E_\alpha\}\) of subspaces \(E_\alpha\subset B\) will be called \(c\)-normalizing
\((c>0)\), if the seminorm
\[ \|x\|_{\mathfrak B}=\sup_{E\in\mathfrak B}\inf_{y\in E}\|x+y\| \tag{3} \]
is equivalent to the original norm of the \(B\)-space and \(\|x\|\geq \|x\|_{\mathfrak B}\geq c\|x\|\) for all \(x\in B\). For us the case of a 1-norming family is especially important; for it \(\|x\|_{\mathfrak B}=\|x\|\). Let us note that (3) is conveniently understood in the following way: let \(P_E x\) denote the image of \(x\) in the quotient space \(B/E\), whose norm we denote by \(\|\cdot\|_E\); then
\[ \|x\|_{\mathfrak B}=\sup_{E\in\mathfrak B}\|P_E x\|_E . \tag{3'} \]
From (3′) it is clear that the notion of a norming family \(\mathfrak B\) contains as a special case the notion of a norming family of functionals, which was widely used by M. I. Kadets, and is its generalization.
Remark 1. Let \(X=\{x_k\}_1^\infty\) be a minimal system in \(B\) with a 1-norming adjoint system \(\{f_k\}_1^\infty\subset B^*\), i.e. \(F=E(\{f_k\})\) is a 1-norming subspace in \(B^*\): \(\sup\{|f(x)|: f\in F\}=\|x\|\) for any \(x\in B\). Denote
\[
\mathfrak B(X)=\{E(\{x_k\}_{k=n+1}^\infty)\}_1^\infty.
\]
Then \(\|x\|_{\mathfrak B}=\|x\|\) for all \(x\in B\).
As noted in (3), every separable \(B\)-space has a complete minimal system with a 1-norming adjoint.
Theorem 2. Let \(X=\{x_k\}_1^\infty\) be a stretching basis in \(B\), i.e. the adjoint system \(X^*=\{f_k\}_1^\infty\) is complete in \(B^*\). Then for any \(\varepsilon>0\) there exists \(n_0(\varepsilon)\) such that, for every \(N\), one can choose \(N_1\) in such a way that for all \(y\in E^N=E(\{x_k\}_{k=N_1+1}^\infty)\)
\[
\|y\|\leq (1+\varepsilon)f(y)
\]
for some \(f\in S(E(\{f_k\}_{k=1}^{n_0},\{f_k\}_{k=N+1}^\infty))\).
Proposition 1. Let \(\{x_k\}_1^\infty\subset B\) be a complete minimal sequence with a 1-norming adjoint \(\{f_k\}_1^\infty\subset B^*\). Then in \(B\) one can introduce an equivalent norm \(\|x\|_1\): \(\|x\|\leq \|x\|_1\leq 2\|x\|\), so that for any \(n\) and \(\varepsilon>0\) there is an \(N=N(\varepsilon,n)\) for which, for every \(y\in E^N=\)
\[ =E(\{x_k\}_{k=N+1}^\infty)\quad \sup\{f(y): f\in E(\{f_k\}_{k=n}^\infty),\ \|f\|_1=1\}\geq (1-\varepsilon)\|y\|_1 . \]
4. The relation between the \(\beta\)- and \(\delta\)-functions of a \(B\)-space and its adjoint
The results of this section essentially use the properties of minimal systems indicated above.
Theorem 3. Let \(X=\{x_k\}_1^\infty\) be an orthogonal basis in \(B\) and let \(X^*=\{f_k\}_1^\infty\) be the biorthogonal functionals in \(B^*\) (i.e. \(f_k(x_j)=\delta_{kj}\)), and suppose \(E(X^*)=B^*\). Then
\[ \delta\delta(\varepsilon;B)=\mathcal L_1[\beta\beta(\eta;B^*)] =\sup_{\eta\geq 0}\frac{\xi\eta-\beta\beta(\eta,B^*)}{1+\beta\beta(\eta,B^*)}. \tag{4} \]
The conditions under which Theorem 3 is true can be weakened considerably; however, because of their cumbersome form we do not dwell on this.
Denote
\[
\sup\{f(\varepsilon;x): x\in S(B)\}=sf(\varepsilon;B)
\]
and
\[
\inf\{f(\varepsilon;x): x\in S(B)\}=if(\varepsilon;B).
\]
Theorem 4. Let \(B\) be a reflexive separable space. Then
\[ s\delta(\varepsilon;B)\leq \mathcal L_1[i\beta(\eta;B^*)] =\sup_{\eta\geq 0}\frac{\xi\eta-i\beta(\eta,B^*)}{1+i\beta(\eta,B^*)} \leq s\delta(2\varepsilon;B). \]
Let us note that if in \(B\) one introduces the equivalent norm indicated in Proposition 1, then the functions \(s\delta(\varepsilon)\) and \(i\beta(\eta)\) will be related by equality (4).
We also note the following proposition, based on Theorem 2.
Proposition 2. Let there exist a stretching basis in \(B\). Then
\[ s\delta(\varepsilon;B)\leq \mathcal L_1[i\beta(\eta,B^*)]. \tag{5} \]
Relations (4) and (5) connect the functions \(\beta(\varepsilon)\) and \(\delta(\varepsilon)\). In order to express the function \(\beta(\varepsilon)\) in terms of \(\delta(\varepsilon)\), we shall need the properties of the transformation (2) indicated in Theorem 1. The function \(\beta\beta(\varepsilon;B)\) is nondecreasing
and \([\beta\beta(\varepsilon;B)+1]/\varepsilon\) does not increase*, but, generally speaking, one cannot assert that the function \(\beta\beta(\varepsilon;B)\) is convex; therefore formulas (4) and (5) should be treated by applying part c) of Theorem 1. Let us illustrate this by the following example.
Corollary 1. Let \(B^*\) be separable and, for some \(c>0\),
\[ \delta\delta(\varepsilon;B) \ge c\varepsilon . \]
Then \(\beta\beta(\eta;B^*)=0\) for \(\eta\le c\). If
\[ \delta\delta(\varepsilon;B^*) \ge c\varepsilon, \]
then \(\beta\beta(\eta;B)=0\) for \(\eta\le c\).
5. On one criterion for complemented subspaces
A basic sequence \(\{x_k\}_1^\infty\) is called boundedly complete (see, for example, \((^6)\), p. 119) if from
\[ \left\|\sum_1^n a_kx_k\right\|<M \]
for all \(n\) there follows the convergence of the series
\[ \sum_1^\infty a_kx_k . \]
Let us note that in a reflexive space every basic sequence is boundedly complete.
Theorem 5. Let \(\{x_k\}_1^\infty\) be a boundedly complete basic sequence in \(B\), \(E(\{x_k\}_1^\infty)=B_1\subset B\). For the complementability of \(B_1\) in \(B\) it is necessary and sufficient that there exist a biorthogonal system \(\{f_k\}_1^\infty\subset B^*\) \((f_k(x_j)=\delta_{kj})\) such that \(B_2=E(\{f_k\}_1^\infty)\) and \(B_1\) are mutually norming subspaces.
Remark 2. If \(\{f_k\}_1^\infty\) is a basic sequence or if \(B_1\) is a reflexive subspace, then it is sufficient to require that \(B_2\) be norming over \(B_1\).
Corollary 2. Denote by \(\{\varphi_k\}_1^\infty\subset B_1^*\) the biorthogonal system to \(\{x_k\}_1^\infty\) \((\varphi_k(x_j)=\delta_{kj})\). If \(\{\varphi_k\}_1^\infty \simeq \{f_k\}_1^\infty\), then \(B_1\) is complemented in \(B\).
It is known \((^4)\) that for \(p\ge2\) every subspace \(E\subset L_p\) and \(E\simeq l_2\) is complemented in \(L_p\). For \(4/3<p<2\) this is not so (see \((^5)\)); however, from Corollary 2 we easily obtain:
Proposition 3. If \(l_2\simeq E\subset L_p\) \((1<p<2)\), then there exists an infinite-dimensional subspace \(E_1\subset E\), and \(E_1\) is complemented in \(L_p\).
Let us also note that a consequence of Theorem 5 is Alaoglu’s theorem (see \((^6)\), p. 120): if \(B\) has a boundedly complete basis, then \(B\) is isomorphic to a conjugate space.
6. Some applications of the duality theorems
It is easy to see that for James’ space (for the definition see \((^6)\), p. 123) \(J\),
\[ \beta^0(\varepsilon;x,J)=\delta^0(\varepsilon;x,J)=\sqrt{1+\varepsilon^2}-1 . \]
It follows from Theorem 3 that in the conjugate space \(J^*\),
\[ \beta^0(\varepsilon;x,J^*)=\delta^0(\varepsilon;x,J^*)=\sqrt{1+\varepsilon^2}-1 . \]
But then (see \((^1)\)) any subspaces \(E_1\subset J\) and \(E_2\subset J^*\) \((\dim E_i=\infty)\) contain \(E_{1,1}\subset E_1\) and \(E_{2,1}\subset E_2\), \(E_{1,1}\simeq l_2\simeq E_{2,1}\). Analogous reasoning with sequences and the use of Corollary 2 shows that: every \(E\subset J\) \((\dim E=\infty)\) contains \(E_1\subset E\) \((\dim E_1=\infty)\), and \(E_1\) is complemented in \(J\).
At the same time I think that the following stronger proposition holds:
Hypothesis. Every subspace \(E\subset J\) is complemented in \(J\).
We now give two results in which the use of Corollary 1 makes it possible to strengthen the corresponding theorems from \((^2)\).
Theorem 6. If \(B\) has an unconditional basis and
\[ \delta^0(\varepsilon;x,B)\ge c\varepsilon\quad (c>0), \]
then there exists \(B_1\subset B\), and \(B_1\simeq l_1\).
Theorem 7. If
\[ \delta^0(\varepsilon;x,B)\ge c\varepsilon\quad (c>0), \]
then \(B\) is not reflexive and in \(B\) there does not exist a shrinking basis \(\{x_k\}_1^\infty\) (i.e., one such that \(E(\{f_k\}_1^\infty)=B^*\), where \(f_k(x_j)=\delta_{kj}\)).
Institute of Chemical Physics
Academy of Sciences of the USSR
Received
20 X 1968
References
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- V. D. Milman, DAN, 179, No. 4, 779 (1968).
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- M. I. Kadec, A. Pełczyński, Studia Math., 21, 161 (1962).
- H. P. Rosenthal, Doct. diss. Stanford Univ., 1965.
- M. M. Day, Normed Linear Spaces, Moscow, 1961.
* The same is true also for the function \(i\beta(\varepsilon;B)\).