Yu. I. Petunin

A Criterion for the Reflexivity of a Banach Space

(Presented by Academician P. S. Aleksandrov on 14 III 1961)

The terminology and notation of this article coincide with the terminology adopted in \((^1)\).

Let \(M'\) denote a vector subspace of the space \(E'\), conjugate to some Banach space \(E\), which is everywhere dense in the weak topology \(\sigma(E', E)\). As is known \((^{1,2})\), the characteristic of the subspace \(M'\) is the least upper bound of the numbers \(t\) such that the weak closure of the intersection of \(M'\) with the unit ball \(S'_1 \subset E'\) contains the ball \(S'_t\) of radius \(t\).

Suppose that on a Banach space \(E\) there is given a certain separable locally convex topology \(T\), which is majorized by the original topology of the space \(E\). Then the space \(E'_T\), conjugate to the space \(E\) endowed with the topology \(T\), will obviously be a vector subspace of \(E'\).

The subspace \(E'_T\) is total, i.e., everywhere dense in the weak topology \(\sigma(E', E)\). Indeed, since the topology \(T\) is separable and locally convex, for every element \(x_0 \in E\) there exists a seminorm \(p(x)\) from the set of seminorms defining the topology \(T\) such that \(p(x_0) \ne 0\). From the Hahn—Banach theorem on the extension of a linear functional in a separable locally convex space it follows that there exists a linear functional \(x'_T\), continuous in the topology \(T\), for which \(\langle x_0, x'_T\rangle = p(x_0) \ne 0\). Hence it follows that \(E'_T\) is total in \(E'\).

Lemma 1. In order that the unit ball \(S_1\) \((\|x\| \le 1)\) of the space \(E\) be closed in the topology \(T\), it is necessary and sufficient that the characteristic of the subspace \(E'_T \subset E'\) be equal to one.

Proof. If \(r\) denotes the characteristic of the subspace \(E'_T \subset E'\), then \(1/r\) is equal to the least upper bound of the numbers \(\|x\|\), where \(x\) runs through the closure of the ball \(S_1\) in \(E\) in the topology \(\sigma(E, E'_T)\) \((^{1,2})\). The proof of Lemma 1 is now obtained from the statement that the closure of a convex set is the same in all separable locally convex topologies on \(E\) consistent with the duality between \(E\) and \(E'_T\).

Theorem 1. In order that a Banach space \(E\) be reflexive, it is necessary and sufficient that its unit ball be a closed set in every separable locally convex topology \(T\) given on \(E\) and comparable with the original topology of the Banach space \(E\).

The proof of the necessity of this assertion is contained in \((^3)\) (see the proof of Lemma 3.3). Moreover, it is obtained at once from Lemma 1 if one takes into account the fact that \(E'_T\), being a dense subspace of \(E'\) in the topology \(\sigma(E', E)\), will be dense in \(E'\) also in the topology \(\sigma(E', E'')\), since for a reflexive space these two topologies coincide. But a vector subspace of \(E'\) dense in the topology \(\sigma(E', E'')\) will be a dense subspace also in the strong topology of the space \(E'\), which means that the characteristic of \(E'_T\) is equal to one.

To prove sufficiency we shall use the following formula for computing the characteristic of the vector subspace \(E_T' \subset E'\):

\[ r=\inf \frac{\|x+z''\|}{\|x\|}, \tag{1} \]

where \(z''\) ranges over \((E_T')^0\), and \(x\) over the set of all nonzero points of \(E\) \((^{1,2})\).

Suppose that \(E\) is nonreflexive. It is not hard to see that in this case there exist elements \(x \in E\) (\(\|x\|=1\)) and \(z'' \notin E\) in \(E''\) such that \(x\) is not orthogonal to the subspace \(\{\lambda z''\}\). This means that

\[ 1>\inf_{\substack{z_1''=\lambda z''\\ x\in E,\ \|x\|=1}} \|z_1''-x\| = \inf_{\substack{z_1''-\lambda z''\\ x\in E}} \frac{\|z_1''-x\|}{\|x\|} =r. \tag{2} \]

Denote by \(M'\) the polar of \(z''\) in the space \(E'\). From (1) and inequality (2) it follows that \(M'\) has characteristic less than one. But \(z'' \notin E\); therefore \(M'\) is a vector subspace of \(E'\), everywhere dense in the weak topology \(\sigma(E',E)\), and hence it follows that \(M'\) and \(E\) are put into duality by the bilinear form \(B(x,x')=\langle x,x'\rangle\), where \(x\in E\), \(x'\in E'\). Therefore \(\sigma(E,M')\) is a separated locally convex topology which, obviously, is majorized by the original topology of the Banach space \(E\). The unit ball \(S_1\) of the space \(E\) is not a closed set in this topology, since its closure contains elements with norm arbitrarily close to \(1/r>1\). The theorem is proved.

The Mackey topology \(\tau(E,M')\) is not normable. However, the following is true:

Lemma 2. If \(E\) is a separable Banach space and \(M'\) is a strongly closed vector subspace of \(E'\), everywhere dense in the weak topology \(\sigma(E',E)\), then the Mackey topology \(\tau(E,M')\) majorizes some normed topology defined on the space \(E\).

Proof. In the space \(E'\), conjugate to a separable Banach space and endowed with the weak topology \(\sigma(E',E)\), the unit ball \(S_1'\) is a separable metrizable space.

The set \(S_1'\cap M'\), being a metrizable space in the topology \(\sigma(M',E)\), is separable in this topology. Let \(e_1',\ldots,e_n',\ldots\) be a total countable set of elements from \(S_1'\cap M'\) in the topology \(\sigma(M',E)\). It is not hard to construct a balanced convex set \(W\subset M'\) which is compact in the strong topology of the space \(E'\) and absorbs each of the one-point sets \(e_n'\). \(W\) will be compact in the weak topology \(\sigma(E',E)\); therefore the polar \(W^0\) in the space \(E\) will belong to the fundamental system of neighborhoods of zero for the Mackey topology \(\tau(E,M')\).

The set \(W\) is total in \(M'\) and, consequently, in \(E'\) in the topology \(\sigma(E',E)\). Hence it follows that \(W^0\) contains no linear manifold. Taking now \(W^0\) as the unit ball of a normed topology defined on \(E\), we obtain the required normed topology.

From Theorem 1 and Lemma 2 it follows:

Theorem 2. For a Banach space \(E\) to be reflexive, it is necessary and sufficient that its unit ball be closed in every normable topology defined on \(E\) and comparable with the original topology of the Banach space \(E\).

Proof. The assertion of the theorem for a separable Banach space \(E\) follows immediately from Theorem 1 and Lemma 2.

Let now \(E\) be a nonseparable nonreflexive Banach space. Then one can find in it a separable nonreflexive subspace \(E_1\). In this subspace, by virtue of the preceding, one can introduce a new norm \(\|x\|_1\), smaller than the original one, such that the ball \(S_1\cap E_1\) is not closed in this norm.

Denote by \(U_1\) the set of all elements of \(E_1\) such that \(\|x\|_1 \leqslant 1\), and by \(U\) the convex hull \(U_1 \cup S_1\). Then the gauge function \(p(x)\) constructed from the set \(U\) gives rise in the space \(E\) to a norm smaller than the original one and coinciding with the norm \(\|x\|_1\) on the subspace \(E_1\). In the norm \(p(x)\) the ball \(S_1\) is not closed. The theorem is proved.

Using the results of (4), Theorem 2 can be formulated in the following form:

Theorem 3. In order that a Banach space \(E\) be related to every Banach space \(E_0\) into which it is normally embedded, it is necessary and sufficient that the space \(E\) be reflexive.

In conclusion, the author expresses his gratitude to S. G. Krein and M. A. Krasnosel’skii for many useful suggestions.

Voronezh State
University

Received
9 III 1961

References

1. N. Bourbaki, Topological Vector Spaces, Moscow, 1959.
2. J. Dixmier, Duke Math. J., 15(4), 1057 (1948).
3. J. L. Massera, J. J. Schäffer, Ann. of Math., 67(2), 517 (1958).
4. S. G. Krein, Yu. I. Petunin, DAN, 139, No. 6 (1961).