D. P. Milman, V. D. Milman

SOME GEOMETRIC PROPERTIES OF NONREFLEXIVE SPACES

(Presented by Academician S. L. Sobolev on 21 III 1963)

1. Two convex cones \(K' \subset B_1\) and \(K'' \subset B_2\), where \(B_1\) and \(B_2\) are Banach spaces, will be called locally isomorphic if between them one can establish a one-to-one linear correspondence \(\varphi\): \(\varphi(K') = K''\), under which the relations
\[ \lim_{n\to\infty}\|x'_n-x'_0\|_1=0 \quad\text{and}\quad \lim_{n\to\infty}\|\varphi(x'_n)-x''_0\|_2=0 \]
hold only jointly, and, moreover, if \(x'_0\in K'\), then \(x''_0=\varphi(x'_0)\in K''\), and conversely.

Let us note that from the local isomorphism of \(K'\) and \(K''\) it does not follow that the relations
\[ \lim_{n\to\infty}\|x'_n-y'_n\|_1=0 \quad\text{and}\quad \lim_{n\to\infty}\|\varphi(x'_n)-\varphi(y'_n)\|_2=0 \]
hold only jointly. Indeed, if the latter relations hold for arbitrary sequences \(x'_n,y'_n\in K'\), then, as is easy to see, the closed linear hulls of the cones \(K'\) and \(K''\) are isomorphic as Banach spaces; however, a local isomorphism of cones, as will be seen from what follows, does not imply, in general, isomorphism of their closed linear hulls.

In what follows, \(\{e_n\}_{1\le n<\infty}\) denotes the natural basis in the space \(l_1\) (of absolutely convergent sequences of numbers), and \(K_l\) is the smallest convex closed cone containing \(\{e_n\}_{1\le n<\infty}\).

Theorem 1. 1) In order that a Banach space \(B\) be nonreflexive, it is necessary and sufficient that it contain a cone \(K\) locally isomorphic to \(K_l\).

2) The indicated cone \(K\) can be chosen so that, in addition, the sequence \(\{\varphi(e_n)\}_{1\le n<\infty}\subset B\), where \(\varphi\) denotes the correspondence establishing the local isomorphism \(\varphi(K_l)=K\), has a biorthogonal sequence of linear functionals.

The sequence \(\{x_n\}_{1\le n<\infty}\), \(x_n=\varphi(e_n)\), will be called below the \(l\)-basis of the cone \(K\).

Remark to Theorem 1. Using Theorem 2 of A. Pełczyński’s paper \((^3)\), one can prove that the cone \(K\) in part 2) of our theorem 1 can be chosen so that the \(l\)-basis \(\{x_n\}\) in it is a basic sequence (i.e., a basis in its closed linear hull).

For an illustration of Theorem 1, let us give an example of a cone \(K\), locally isomorphic to \(K_l\), in the space \(c_0\) (of sequences of numbers converging to zero). It suffices to specify an \(l\)-basis of such a cone. It is the sequence \(\{x_n\}_{1\le n<\infty}\), where \(x_n\) has its first \(n\) coordinates equal to 1, and the remaining ones equal to zero. One can show that the sequence \(\{x_n\}_{1\le n<\infty}\) in this example is also a conditional basis in the space \(c_0\).

Relying on Theorem 1, one can establish the following result:

Theorem 2. For the reflexivity of a Banach space \(B\), it is necessary and sufficient that every affine continuous mapping of an arbitrary nonempty convex closed bounded set onto itself have a fixed point.

2. The results of the preceding section use the Šmulian–Eberlein theorem \((^{1,2})\), according to which a necessary and sufficient condition

of nonreflexivity of a Banach space \(B\) is the existence in it of a countable system of nonempty convex closed bounded sets \(\{G_n\}_{1<n<\infty}\), \(G_{n+1}\subset G_n\), having empty intersection. In what follows we shall call such a system a deposit with empty intersection and write \(\pi=\{G_n\}\). By studying such deposits one discovers a number of geometric properties of nonreflexive Banach spaces, some of which are set out below.

We shall call a deposit \(\pi_1=\{G_n^1\}\) subordinate to a deposit \(\pi=\{G_n\}\) and write \(\pi_1\prec\pi\), if \(G_n^1\subset G_n\) for all \(n\), \(1\le n<\infty\).

Denote by \(r(x,G_n)\) the distance from the point \(x\) to the set \(G_n\); \(r(x,\pi)=\lim_{n\to\infty} r(x,G_n)\); \(r(\pi)=\lim_{n\to\infty}\inf_{x\in G_n} r(x,\pi)\)*. One can show that the number \(r(\pi)\) is the exact lower bound of the numbers

\[ \lim_{n\to\infty}\lim_{m\to\infty}\|x^n-y^{n+m}\|, \]

where \(\{x^n,y^n\in G_n\}\). It is obvious that for a deposit \(\pi_1\prec\pi\) one has \(r(\pi_1)\ge r(\pi)\).

A deposit \(\pi=\{G_n\}\) with empty intersection will be called \(\omega\)-split if \(r(\pi)>0\).

A bounded sequence of elements \(\{u_n\}_{1<n<\infty}\) of the space \(B\) will be called \(\omega\)-split if there exists \(\delta>0\) such that
\(\delta\le \lim_{m\to\infty}\|u_{m n_m}-u_{n_m\omega}\|\), where \(u_{m n}\) and \(u_{n\omega}\) denote arbitrary elements from the convex hulls of the elements \(\{u_j\}_{m<j\le n}\) and \(\{u_j\}_{n<j<\infty}\), respectively, and \(n_m\) is an arbitrary number greater than \(m\). The exact upper bound of such numbers \(\delta\), taken over all subsequences \(\{u_{n_k}\}_{1\le k<\infty}\), will be called the index of \(\omega\)-splitness of the sequence \(\{u_n\}_{1\le n<\infty}\).

Theorem 3. 1) For every deposit \(\pi\) with empty intersection there is a subordinate deposit \(\pi_0\prec\pi\) which is \(\omega\)-split; every nonreflexive space contains an \(\omega\)-split deposit. 2) Let the deposit \(\pi_0=\{G_n^0\}\) be \(\omega\)-split; then from every sequence \(\{x^n\in G_n^0\}_{1\le n<\infty}\) one can extract a subsequence \(\{u_n\}_{1\le n<\infty}\) which is \(\omega\)-split. 3) In order that the space \(B\) be nonreflexive it is necessary and sufficient that it contain an \(\omega\)-split sequence.

Remark. Let \(\pi=\{G_n\}\). The number \(r(\pi)\) is equal to the exact lower bound of the indices of \(\omega\)-splitness over all sequences \(\{x_n\}\), \(x_n\in G_n\).

1. Denote by \(d(M)\) the diameter of the set \(M\subset B\). The number \(d(\pi)=\lim_{n\to\infty} d(G_n)\) will be called the diameter of the deposit \(\pi=\{G_n\}\). It is obvious that if \(\pi_1\prec\pi\), then \(d(\pi_1)\le d(\pi)\).

A deposit \(\pi=\{G_n\}\) will be called \(\omega\)-diametral if for any elements \(y_n,x_n\in G_n\) one has

\[ \lim_{n\to\infty}\lim_{m\to\infty}\|x_n-y_{n+m}\|=d(\pi)>0. \]

A bounded sequence of elements \(\{u_n\}_{1\le n<\infty}\) of the space \(B\) will be called \(\omega\)-diametral if for any elements \(u_{m n}\) from the convex hull of \(\{u_j\}_{m<j\le n}\) and \(u_{n\omega}\) from the convex hull of \(\{u_j\}_{n<j<\infty}\) one has
\(\lim_{m\to\infty}\|u_{m n_m}-u_{n_m\omega}\|=\lim_{n\to\infty} d(\{u_j\}_{n<j<\infty})\), where \(n_m\) is an arbitrary

* One can give an example of a deposit for which

\[ \lim_{n\to\infty}\inf_{x\in G_n} r(x,\pi)\ne \inf_{x\in G_1} r(x,\pi). \]

number greater than \(m\), and \(d(\{u_j\}_{n<j})\) denotes the diameter of the sequence \(\{u_j\}_{n<j<\infty}\).

It can be proved that a nesting \(\pi=\{G_n\}\) is \(\omega\)-diametral if and only if two conditions are simultaneously satisfied: a) \(d(\pi)=r(\pi)\); b) for any point \(x\in G_1\) one has
\[ r(x,\pi)=\lim_{n\to\infty} r(x,y_n) \]
for an arbitrary sequence \(\{y_n\}\), \(y_n\in G_n\), \(1\le n<\infty\),—the property of “compactness of the nesting” with respect to any point of the set \(G_1\).

Theorem 4. 1) For every nesting \(\pi\) with empty intersection there is an \(\omega\)-diametral nesting \(\pi_0<\pi\) subordinate to it; every nonreflexive space contains an \(\omega\)-diametral nesting. 2) Let the nesting \(\pi_0=\{G_n\}\) be \(\omega\)-diametral; then from any sequence \(\{x_n\}_{1\le n<\infty}\), \(x_n\in G_n\), one can choose a subsequence that is \(\omega\)-diametral. 3) Every nonreflexive Banach space contains a cone \(K\), locally isomorphic to the cone \(K_l\), and moreover such that its \(l\)-basis is an \(\omega\)-diametral sequence and a basis in its closed linear span.

An example of an \(\omega\)-diametral sequence is the natural basis \(\{e_n\}_{1\le n<\infty}\) of the space \(l_1\). In the space \(c_0\) (sequences of numbers converging to zero) an example of an \(\omega\)-diametral sequence is the sequence \(\{x_n\}_{1\le n<\infty}\), where \(x_n\) has its first \(n\) coordinates equal to 1, and the remaining ones equal to zero.

A consequence of Theorem 4 is the result formulated below, which is directly connected with the fact \({}^{4}\) that every uniformly convex Banach space is reflexive.

We shall call an \(n\)-dimensional simplex, one of whose vertices is the origin, \(\varepsilon\)-directionally normalized \((1>\varepsilon\ge 0)\) if there exists a numbering of its vertices \(\{z_k\}_{0\le k\le n}\), \(z_0=0\), such that for any \(j\), \(0\le n<j\), one has
\[ 1\ge \|x_j-y_j\|\ge 1-\varepsilon, \]
where \(x_j\) is any element of the convex hull of \(\{z_k\}_{0\le k\le j}\), and \(y_j\) is any element of the convex hull of \(\{z_k\}_{j<k\le n}\).

Corollary to Theorem 4. If the space \(B\) is nonreflexive, then for every \(\varepsilon>0\) it contains an \(\varepsilon\)-directionally normalized simplex of arbitrarily high dimension.

Odessa Electrotechnical Institute of Communications
Physico-Technical Institute of Low Temperatures
of the Academy of Sciences of the Ukrainian SSR

Received
14 III 1963

References

1. V. L. Shmulian, Mat. sborn., 5 (47), 317 (1939).
2. W. A. Eberlein, Proc. Nat. Acad. Sci. USA, 33, 51 (1947).
3. A. Pełczyński, Studia Math., 21, 371 (1962).
4. D. P. Milman, DAN, 20, 243 (1938).