Yu. I. Petunin

DUAL BANACH SPACES CONTAINING SUBSPACES OF CHARACTERISTIC ZERO

(Presented by Academician A. I. Mal’tsev on 2 IX 1963)

Let us consider a Banach space \(E\) and its dual space \(E'\). Denote by \(M'\) a vector subspace of the space \(E'\), everywhere dense in the weak topology \(\sigma(E',E)\), and by \(\|x\|^*\) the norm of an element \(x\in E\) generated by the functionals from the subspace \(M'\):

\[ \|x\|^*=\sup_{\substack{x'\ne 0,\ x'\in M'}} \frac{|\langle x,x'\rangle|}{\|x'\|}. \tag{1} \]

The subspace \(M'\) is called a subspace with zero characteristic if the norm \(\|x\|^*\) is not equivalent to the original norm \(\|x\|\), defined in the space \(E\) (see \((^2)\)).

In \((^5)\) it was shown that, in the case of a reflexive space \(E\), the norm (1) always coincides with the original norm \(\|x\|\), so that the question of the existence of subspaces \(M'\) with zero characteristic is naturally posed only for nonreflexive Banach spaces.

Theorem 1. If a Banach space \(E\) is isomorphic to the Cartesian product \(E\times F\), where \(F\) is some nonreflexive Banach space, then the dual space \(E'\) contains a subspace \(M'\), everywhere dense in the weak topology \(\sigma(E',E)\), whose characteristic is zero.

Lemma 1. Let \(E\) be a Banach space containing a separable subspace \(E_1\) such that in the dual space \(E_1'\) there exists a subspace of characteristic zero, everywhere dense in the topology \(\sigma(E_1',E_1)\). Then the dual space \(E'\) also contains a subspace of characteristic zero, everywhere dense in the topology \(\sigma(E',E)\).

Lemma 2. The lower bound of two separate locally convex topologies \(\mathfrak T\) and \(\mathfrak T_1\), given on the vector space \(E\), in the set of all topologies compatible with the structure of the vector space \(E\), is a locally convex topology.

Proof. Let \(T_1\) be the lower bound of the topologies \(\mathfrak T\) and \(\mathfrak T_1\) in the set of all locally convex topologies, and let \(T\) be the lower bound of \(\mathfrak T\) and \(\mathfrak T_1\) in the set of all topologies compatible with the structure of the vector space \(E\).

Denote by \(\mathfrak B\) a fundamental system of neighborhoods of zero of the topology \(T_1\), whose existence is asserted in \((^1)\) (see Proposition 5, p. 27). Consider an arbitrary neighborhood \(V\in\mathfrak B\), and let \(W\) be a neighborhood of zero from \(\mathfrak B\) such that \(W+W\subset V\). The neighborhood \(W\) contains convex neighborhoods of zero \(\mathcal U\) and \(\mathcal U_1\) in the topologies \(\mathfrak T\) and \(\mathfrak T_1\), respectively, since the topology \(T_1\) is weaker than the topologies \(\mathfrak T\) and \(\mathfrak T_1\). It is easy to see that the convex hull \(K\) of the set \(\mathcal U\cup\mathcal U_1\subset W\) belongs to the set \(W+W\). Indeed, if \(z\in K\), then \(z=tx+(1-t)x_1\) \((0\le t\le 1)\), where \(x\in\mathcal U\), \(x_1\in\mathcal U_1\); but \(tx\in\mathcal U\subset W\), \((1-t)x_1\in\mathcal U_1\subset W\), since \(\mathcal U\) and \(\mathcal U_1\) are convex sets, so that \(tx+(1-t)x_1\in W+W\), and the inclusion \(K\subset W+W\) is proved.

It remains to note that the sets \(K\) form a fundamental system of neighborhoods of zero of the topology \(T\), when \(\mathcal U\) and \(\mathcal U_1\) run through fundamental systems of neighborhoods of zero in the topologies \(\mathfrak T\) and \(\mathfrak T_1\). Lemma 2 is proved.

Proof of Theorem 1. It follows from the hypotheses of the theorem that in the Banach space \(E\) one can select a sequence of nonreflexive closed subspaces \(F_1, F_2, \ldots, F_n, \ldots\), possessing the following properties:

1. \(F_i \cap F_j=\theta\), if \(i\ne j\).
2. For every natural \(n\) the direct sum \(E_n=F_1+F_2+\ldots+F_n\) is closed in the space \(E\).

Without loss of generality we may assume that each of the subspaces \(F_1, F_2, \ldots, F_n, \ldots\) is separable. From the results of paper \((^5)\) it follows that on the subspace \(F_n\) one can define a new normed topology \(T_{(n)}\), weaker than the original topology of the space \(F_n\), in such a way that the closure of the unit ball \(S_1^{(n)}\) of the space \(F_n\) in the topology \(T_{(n)}\) contains elements \(x\) whose norms \(\|x\|\) exceed the number \(n\).

Consider on the subspace \(E_n=F_1+F_2+\ldots+F_n\) the direct sum \(P_n\) of the topologies \(T_{(1)}, T_{(2)}, \ldots, T_{(n)}\), and denote by \(\widetilde E\) the union of all subspaces \(E_n\). It is easy to see that, for every \(n\), the topology induced on \(E_n\) by the topology \(P_{n+1}\) coincides with the topology \(P_n\). We may therefore define on the vector space \(\widetilde E\) the strict inductive limit \(\mathscr T\) of the normed topologies \(P_n\).

We shall further denote by \(\mathscr T_1\) the topology induced on \(\widetilde E\) by the original topology of the space \(E\), and by \(T_1\) the lower bound of the topologies \(\mathscr T\) and \(\mathscr T_1\) in the set of all locally convex topologies. It is not hard to note that \(T_1\) is a separated locally convex topology on the space \(\widetilde E\).

If \(\mathcal U\) and \(\mathcal U_1\) are two neighborhoods from the fundamental system of neighborhoods of zero in the topologies \(\mathscr T\) and \(\mathscr T_1\), then the convex hull \(K\) of the set \(\mathcal U\cup \mathcal U_1\) will form a fundamental system of neighborhoods of zero of the topology \(T_1\), as \(\mathcal U\) and \(\mathcal U_1\) range over the set of all neighborhoods from the fundamental system of neighborhoods of zero of the topologies \(\mathscr T\) and \(\mathscr T_1\).

The topology \(T_1\) is the upper bound for the set of all locally convex topologies that are weaker than the topology \(\mathscr T_1\) of the space \(\widetilde E\) and induce on each of the subspaces \(E_n\) the topology \(P_n\).

The closure of the unit ball \(\widetilde S=\widetilde E\cap S\), where \(S\) is the unit ball of the space \(E\), in the topology \(T_1\) is an unbounded set in the norm of the space \(E\). If we prove that the completion of the vector space \(\widetilde E\), endowed with the topology \(T_1\), contains the closure \(\widetilde E_1\) of the subspace \(\widetilde E\) in the Banach space \(E\), then it will follow that in the conjugate space \(\widetilde E_1'\) there exists an everywhere dense, in the topology \(\sigma(\widetilde E_1',\widetilde E_1)\), subspace of characteristic zero. For this purpose consider the lower bound \(T\) of the topologies \(\mathscr T\) and \(\mathscr T_1\) in the set of all topologies compatible with the structure of the vector space \(\widetilde E\), and show that the completion \(\widetilde E\), endowed with the topology \(T\), contains \(\widetilde E_1\).

The latter condition is equivalent to the following assertion: every sequence \((x_n)\) fundamental in the norm \(\mathscr T_1\) and converging to zero in the topology \(T\) tends to zero in the topology \(\mathscr T_1\). It is easy to see that every net \((z_n)\) converging to zero in the topology \(T\) has the form \(z_n=x_n+y_n\), where \(x_n\) and \(y_n\) are nets converging to zero in the topologies \(\mathscr T\) and \(\mathscr T_1\), respectively. If the sequence \((x_n)\) converges to zero in the topology \(\mathscr T\), then it belongs to one of the subspaces \(E_n\). To prove this, note that each of the subspaces \(E_n\) is closed in \(E_{n+1}\) (in the topology \(P_{n+1}\)); therefore \(E_n\) is also closed in the topology \(\mathscr T\) (see \((^1)\), Remark 2, p. 88). The sequence \((x_n)\) is a bounded set in the topology \(\mathscr T\), since \((x_n)\) is a fundamental sequence. By the known theorem on bounded sets in a strict inductive limit (see \((^1)\), Proposition 6, p. 151), we may assert that the sequence \((x_n)\) is contained in one of the subspaces \(E_n\).

The space \(E_n\), endowed with the induced topology \(\mathscr T_1\), is ...

Banach space; therefore the sequence \((x_n)\) converges to zero in the norm \(\mathcal{T}_1\). Hence it follows that the sequence \(z_n=x_n+y_n\) converges to zero in the topology \(\mathcal{T}_1\), so that the completion of the space \(\widetilde E\), endowed with the topology \(T\), contains the Banach space \(\widetilde E_1\).

From Lemma 2 it follows that the topologies \(T\) and \(T_1\) are equivalent. Thus, on the space \(\widetilde E_1\) there is defined a separable locally convex topology \(T_1\), weaker than the original topology of the space \(\widetilde E_1\), and the closure of the unit ball \(\widetilde S_1\) of the space \(\widetilde E_1\) in the topology \(T_1\) is unbounded in the norm of \(\widetilde E_1\). Consequently, \(\widetilde E_1\) has an everywhere dense, in the weak topology, subspace of characteristic zero, and therefore (Lemma 1) the conjugate space \(E'\) contains an analogous subspace. The theorem is proved.

Corollary 1. If a nonreflexive Banach space \(E\) is isomorphic to its Cartesian square \(E\times E\), then the conjugate space \(E'\) contains an everywhere dense, in the topology \(\sigma(E',E)\), subspace of characteristic zero.

With the aid of Theorem 1, Lemma 1, and Theorem 2 from the paper of R. James \((^3)\), it is not difficult to prove

Corollary 2. Let \(E\) be a Banach space that contains a nonreflexive subspace \(E_1\) with an unconditional basis. Then in the conjugate space \(E'\) there exists an everywhere dense, in the topology \(\sigma(E',E)\), subspace \(M'\) whose characteristic is equal to zero.

The results presented make it possible to establish the existence of subspaces with zero characteristic, everywhere dense in the topology \(\sigma(E',E)\), in the conjugate spaces of the Banach spaces \(c_0\), \(c\), \(l_1\), \(m\), \(C(0,1)\), \(L_1(0,1)\), \(M(0,1)\), \(C_\alpha\) (Hölder space), etc.

A Banach space \(E\) is called quasi-reflexive if the quotient space \(E''/E\) is finite-dimensional \((^4)\). It is easy to see that, for a quasi-reflexive Banach space, the conditions of Theorem 1 are not satisfied.

Theorem 2. Every subspace \(M'\) of the conjugate space \(E'\) of a quasi-reflexive Banach space, everywhere dense in the weak topology \(\sigma(E',E)\), has characteristic \(r_{M'}\) different from zero.

Definition 1. We shall say that a Banach space \(E\) has the \(R\)-property if it contains an infinite-dimensional reflexive subspace.

Theorem 3 (criterion of quasi-reflexivity of a separable Banach space). Let \(E\) be a separable Banach space. In order that \(E\) be quasi-reflexive, it is necessary and sufficient that the conjugate space have the \(R\)-property and that the second conjugate space \(E''\) contain no subspaces with zero characteristic, everywhere dense in the topology \(\sigma(E'',E')\).

The question of the existence of subspaces \(M'\subset E'\), everywhere dense in the weak topology \(\sigma(E',E)\) and with zero characteristic in an arbitrary non-quasi-reflexive Banach space, remains open.

The author expresses gratitude to S. G. Krein for his attention and valuable advice.

Voronezh State
University

Received
19 VIII 1963

CITED LITERATURE

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4. P. Civin, B. Yood, Proc. Am. Math. Soc., 8, 906 (1957).
5. Yu. I. Petunin, DAN, 140, No. 1 (1961).