M. I. KADEC

TOPOLOGICAL EQUIVALENCE OF CERTAIN CONES OF A BANACH SPACE

(Presented by Academician L. V. Kantorovich, XII 25, 1964)

Let us call, following M. G. Krein, a cone \(K\) of a Banach space \(E\) normal if there exists \(\varepsilon > 0\) such that for any normalized elements \(x\) and \(y\) of the cone \(\|x+y\| \geq \varepsilon\). Let us call a cone strongly acute if its elements satisfy the condition

\[ \|x+z\| \geq \|x\|+\eta(x,\|z\|)\qquad (x,z\in K;\quad \eta(x,\|z\|)>0\ \text{for}\ \|z\|>0). \tag{1} \]

Theorem 1. In a separable Banach space \(E\) with a normal cone \(K\), one can introduce an equivalent norm with respect to which the cone \(K\) will be strongly acute.

The proof follows immediately from the following two propositions.

Proposition 1. In the space \(C(S)\) of functions continuous on a compact set \(S\), one can introduce an equivalent norm with respect to which the cone of nonnegative functions will be strongly acute.

Proof. Let the diameter of the compact set \(S\) be taken to be equal to one. For each \(\varepsilon > 0\) define a finite \(\varepsilon\)-covering, i.e., a collection of closed sets
\(S_i(\varepsilon)\subset S\) \((i=1,2,\ldots,n(\varepsilon))\) with diameters not exceeding \(\varepsilon\), covering \(S\). For each \(\varepsilon\) define the functional

\[ J(x,\varepsilon)=\frac{1}{n(\varepsilon)} \sum_{i=1}^{n(\varepsilon)} \max_{s\in S_i(\varepsilon)} |x(s)| \qquad (x(s)\in C(S);\ 0<\varepsilon\leq 1) \tag{2} \]

and with its help define the desired norm

\[ \|x\|=\sum_{k=0}^{\infty} 2^{-k} J(x,2^{-k}). \tag{3} \]

It is easy to see that

\[ \max_{s\in S}|x(s)| \leq \|x\| \leq 2\max_{s\in S}|x(s)|. \]

Now consider nonnegative functions \(x(s)\) and \(z(s)\); \(\|x\|=1\), \(\|z\|=\delta\). Choose \(k_0=k_0(x)\) so large that the oscillation of the function \(x(s)\) on each of the sets \(S_i(2^{-k_0})\) does not exceed \(\frac12\delta\). Then on that one of the sets \(S_i(2^{-k_0})\) where \(z(s)\) attains its greatest value,

\[ \max [x(s)+z(s)] \geq \max x(s)+\delta/4 \]

and therefore, according to (2),

\[ J(x+z,2^{-k_0}) \geq J(x,2^{-k_0})+ \frac{1}{n(2^{-k_0})}\cdot\frac{\delta}{4}. \tag{4} \]

From (3) and (4) it follows that

\[ \|x+z\|\geq \|x\|+\frac{2^{-k_0}}{n(2^{-k_0})}\cdot \frac{\delta}{4}, \]

which proves Proposition 1.

Proposition 2. A separable Banach space with a normal cone \(K\) can be embedded isomorphically in the space \(C(S)\), and \(K\) (and only \(K\)) is mapped into the cone of nonnegative functions.

This proposition, generalizing the Banach—Mazur theorem on a universal space, was proved by M. G. Krein \((^1)\).

We now define in the separable Banach space \(E\) a complete minimal system

\[ x_1,x_2,x_3,\ldots \qquad (\|x_i\|=1) \tag{5} \]

with total adjoint \(\{f_i\}_1^\infty\). Consider the smallest closed cone \(K\) spanned by the elements of the system (5). Suppose that it is strongly acute.

Lemma 1. If \(x\in K\), then \(\lim\limits_{n\to\infty} S_n x=x\), where

\[ S_n x=\sum_{k=1}^n f_k(x)x_k . \]

The proof rests on the monotonicity of the norm in \(K\).

Consider the set \(D_0\) of those elements \(x\in K\) for which \(\|x\|\leq 1\). For each \(y\in D_0\) define the set

\[ D(y)=\{z:z-y\in K,\ \|z\|\leq 1\}. \]

Lemma 2. If \(y_k\to y\) \((y_k\in D_0)\), then

\[ \lim_{k\to\infty}\rho(D(y_k),D(y))=0\ *. \]

Proof. The case \(\|y\|<1\) is examined quite simply without using any special properties of the cone \(K\). Let \(\|y\|=1\). Since the cone \(K\) is strongly acute, \(D(y)\) contains only the point \(y\). Let \(z_k\) be an arbitrary point of the set \(D(y_k)\). Then

\[ \lim_{k\to\infty}\|y-y_k+z_k\| \leq \lim_{k\to\infty}\|y-y_k\|+\lim_{k\to\infty}\|z\|=1, \]

and, according to condition (1), \(\lim \|y_k-z_k\|=0\), i.e. the diameter of the set \(D(y_k)\) tends to zero as \(k\to\infty\). Since at the same time the point \(y_k\) itself tends to \(y\), the distance between \(D(y_k)\) and \(D(y)\) tends to zero.

For each normalized element \(x\in D_0\), define the sequence of numbers

\[ \delta_n(x)=\frac{d\{D(S_n x)\}}{d\{D_0\}}\prod_{k=1}^n \left(1-\frac{f_k(x)}{2^k\|f_k\|}\right) \qquad (n=1,2,\ldots), \]

where \(d\{M\}\) is the diameter of the set \(M\). From Lemmas 1—2 it follows that

\[ 1\geq \delta_1(x)\geq \delta_2(x)\geq \cdots;\qquad \lim_{n\to\infty}\delta_n(x)=0. \]

Lemma 3. Let the sequence of numbers \(\Delta_i\) be subject to the conditions

\[ 1\geq \Delta_1\geq \Delta_2\geq\cdots;\qquad \lim_{n\to\infty}\Delta_n=0. \]

Then there exists a unique normalized element \(x\in D_0\) for which
\[ \delta_n(x)=\Delta_n\qquad (n=1,2,\ldots). \]

\[ \text{* }\rho(X,Y)\text{ is the Hausdorff distance between subsets of a metric space.} \]

Proof. Consider the system of equations

\[ d\left\{D\left(\sum_{i=1}^{n}\lambda_i x_i\right)\right\} \prod_{i=1}^{n}\left(1-\frac{\lambda_i}{2^i\|f_i\|}\right) = d\{D_0\}\Delta_n \quad (n=1,2,\ldots). \tag{6} \]

For fixed values \(\lambda_i \ge 0\) \((i<n)\), the left-hand side of the equation is a strictly decreasing function of \(\lambda_n \ge 0\), vanishing when

\[ \left\|\sum_{i=1}^{n}\lambda_i x_i\right\|=1, \]

and taking the value \(d\{D_0\}\Delta_{n-1}\) when \(\lambda_n=0\). From what has been said it follows that the system (6) has a unique solution \(\{\lambda_i\}_1^n\) \((\lambda_n\ge 0)\). The sets

\[ D\left(\sum_{i=1}^{n}\lambda_i x_i\right) \]

form a decreasing sequence, and their diameters decrease without bound as \(n\to\infty\). The unique element

\[ x=\sum_{i=1}^{\infty}\lambda_i x_i, \]

lying in their intersection, will be the desired element.

Theorem 2. A normal cone \(K\), spanned by the vectors of a complete minimal system with total adjoint, is homeomorphic to the cone \(K(l_2)\), spanned by the orthonormal basis of the space \(l_2\).

Proof. By Theorem 1, without loss of generality one may assume that \(K\) is strongly acute. With the aid of the constructions carried out above, to each element \(x\in K\) we assign, one-to-one, the nonincreasing sequence

\[ \Delta_n=\delta_n\left(\frac{x}{\|x\|}\right)\|x\| \quad (n=0,1,2,\ldots;\ \Delta_0=\|x\|), \tag{7} \]

which tends to zero as \(n\to\infty\). The resulting correspondence is also mutually continuous if on the set \(\widetilde K\) of sequences (7) we introduce the topology of coordinatewise convergence. Let us show this. Suppose the sequence \(y_\nu\in K\) converges to \(y\). Without loss of generality one may assume \(\|y_\nu\|=1\). From the convergence of \(y_\nu\) to \(y\) it follows that \(f_n(y_\nu)\to f_n(y)\) and \(S_n y_\nu\to S_n y\) \((n=1,2,\ldots)\), whence, by Lemma 2,

\[ \delta_n(y_\nu)\to\delta_n(y) \quad (n=1,2,\ldots). \tag{8} \]

On the other hand, if (8) holds, then, considering (6) successively for \(n=1,2,\ldots\), we conclude that

\[ \lim_{\nu\to\infty} S_n y_\nu=S_n y \quad (n=1,2,\ldots). \]

Given an arbitrary \(\varepsilon>0\), choose \(n\) so large that \(d\{D(S_n y)\}<\varepsilon/2\). Then choose \(\nu\) so that

\[ \rho\bigl(D(S_n y);D(S_n y_\nu)\bigr)<\varepsilon/2, \]

which can be done on the basis of Lemma 2. Since \(y_\nu\in D(S_n y_\nu)\), and \(y\in D(S_n y)\), it follows from the preceding inequalities that \(\|y_\nu-y\|<\varepsilon\). Thus, every cone \(K\) satisfying the conditions of Theorem 2 is homeomorphic to \(\widetilde K\). Since the cone \(K(l_2)\) satisfies the conditions of Theorem 2, \(K\) is homeomorphic to \(K(l_2)\).

Theorem 3. Let \(E\) be a Banach space with an unconditional basis.\(^*\) Then the cone spanned by the elements of this basis is homeomorphic to \(K(l_2)\).

The proof, according to Theorem 2, follows from the fact that the cone spanned by the elements of an unconditional basis is normal.

Theorem 4. Every Banach space with an unconditional basis is homeomorphic to \(l_2\).

Proof. We first establish a topological correspondence between the cone \(K\), spanned by the elements of the unconditional basis

\(^*\) For the definition and properties of an unconditional basis see \((^3)\).

\(\{x_k\}_1^\infty\) of the space \(E\), and \(K(l_2)\), generated by the orthogonal basis \(\{e_k\}_1^\infty\) of the space \(l_2\). Let \(x=\sum_1^\infty a_k x_k \in E\). If to the element \(x'=\sum a_k' |x_k| \in K\) there corresponds the element \(y'=\sum b_k e_k \in K(l_2)\), then to the element \(x\) we assign the element \(y=\sum b_k \operatorname{sign} a_k \cdot e_k\). This will be the required homeomorphism.

The result of Theorem 4 was established in another way by Cz. Bessaga and A. Pełczyński \((^2)\); in their proof a special case of Theorem 4 is used (a homeomorphism of the spaces \(C\) and \(l\)), obtained earlier by the author \((^4)\). H. Corson and V. Klee \((^5)\) established a homeomorphism between \(l_2\) and \(K(l_2)\); from this result and Theorems 3 and 4 one obtains:

Theorem 5. A space \(E\) with an unconditional basis is homeomorphic to its cone \(K(E)\).

It is unknown whether in Theorems 3–5 one can dispense with the requirement that the basis be unconditional.

Received
16 XII 1964

References Cited

\(^1\) M. G. Kreĭn, DAN, 28, No. 1, 13 (1940).
\(^2\) Cz. Bessaga, A. Pełczyński, Bull. Acad. Polon., 8, No. 11–12, 757 (1960).
\(^3\) M. M. Day, Normed Linear Spaces, Moscow, 1961.
\(^4\) M. I. Kadets, DAN, 92, No. 3, 465 (1953).
\(^5\) H. Corson, V. Klee, Proc. Symp. Pure Math., 7, Convexity, Am. Math. Soc., 1963.