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|\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|\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}).

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}) 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\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) denote arbitrary elements from the convex hulls of the elements ({u_j}}|u_{m n_m}-u_{n_m\omega}|), where (u_{m n}) and (u_{n\omega{m<j\le n}) and ({u_j}}}), 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}).

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}) 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) from the convex hull of ({u_j}{m<j\le n}) and (u) from the convex hull of ({u_j}{n<j<\infty}) one has
(\lim)), where (n_m) is an arbitrary}|u_{m n_m}-u_{n_m\omega}|=\lim_{n\to\infty} d({u_j}_{n<j<\infty

* 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}).

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}), 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}).}), 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).