Reports of the Academy of Sciences of the USSR

1. Vol. 163, No. 5

MATHEMATICS

V. I. GURARII

SPACES OF UNIVERSAL DISPOSITION

(Presented by Academician L. V. Kantorovich on 25 I 1965)

Let \(E\) be a Banach space; \(P\) a subspace in \(E\); \(\mathfrak M\) a certain class of Banach spaces; \(\mathfrak R\) the class of all finite-dimensional Banach spaces; \(\mathfrak H\) the class of all separable Banach spaces.

Definition 1. We shall call \(E\) \(a\)-universal \((a \ge 1)\) with respect to \(\mathfrak M\) if for every \(F \in \mathfrak M\) there is a subspace \(\tilde F\) in \(E\) and an isomorphism \(\varphi\) of \(F\) onto \(\tilde F\) such that
\[ \max\{\|\varphi\|,\|\varphi^{-1}\|\} \le a. \]
If \(E\) is \(a\)-universal with respect to \(\mathfrak M\) for every \(a>1\) (respectively for \(a=1\)), then we shall call \(E\) almost universal (respectively universal) with respect to \(\mathfrak M\).

Thus, for example, each of the spaces \(C, m, A\)* is universal with respect to \(\mathfrak H\) \((^{1,2})\); one can show that \(c\) is almost universal with respect to \(\mathfrak R\).

Definition 2. We shall call \(P\) a subspace of \(a\)-universal \((a \ge 1)\) disposition in \(E\) with respect to \(\mathfrak M\), if for every pair of Banach spaces \(\tilde P, \tilde R\) (\(\tilde P\) is a subspace in \(\tilde R\)), where \(\tilde P\) is isomorphic to \(P\), \(\tilde P \in \mathfrak M\), \(\tilde R \in \mathfrak M\), and an isomorphism \(\varphi\) of \(\tilde P\) onto \(P\), there exists a subspace \(R\) in \(E\), \(R \supset P\), and an isomorphism \(\Phi\) of \(\tilde R\) onto \(R\) such that: 1) \(\Phi=\varphi\) on \(\tilde P\); 2) \(\|\Phi\| \le a\|\varphi\|\), \(\|\Phi^{-1}\| \le a\|\varphi^{-1}\|\). If \(P\) is a subspace of \(a\)-universal disposition in \(E\) with respect to \(\mathfrak M\) for every \(a>1\) (respectively for \(a=1\)), then we shall call \(P\) a subspace of almost universal (respectively universal) disposition in \(E\) with respect to \(\mathfrak M\).

Definition 3. We shall call \(E\) a space of \(a\)-universal \((a \ge 1)\) (respectively almost universal, universal) disposition with respect to \(\mathfrak M\), if every subspace \(P\) in \(E\) is a subspace of \(a\)-universal (respectively almost universal, universal) disposition in \(E\) with respect to \(\mathfrak M\).

The main purpose of our note is to establish the existence of a separable Banach space of almost universal disposition with respect to \(R\) (Theorem 2). In what follows we shall say that \(B_1\) can be embedded in \(B_2\) (\(B_1, B_2\) are Banach spaces) if there exists an isometry of \(B_1\) onto some subspace \(\tilde B_1\) of \(B_2\), and in doing so we shall identify \(B_1\) with \(\tilde B_1\).

Theorem 1. Let \(P_1,E_1\) and \(P_2,E_2\) be two pairs of Banach spaces; \(P_1\) is a subspace in \(E_1\); \(P_2\) is a subspace in \(E_2\), and \(\varphi\) is an isomorphism of \(P_1\) onto \(P_2\). Then \(E_2\) can be embedded in a Banach space \(B\),
\[ \dim B \le \dim E_2 + \dim E_1/P_1 \]
with a subspace \(\tilde E \supset P_2\) such that there exists an isomorphism \(\Phi\) of \(E_1\) onto \(\tilde E\), coinciding with \(\varphi\) on \(P_1\) and satisfying the condition
\[ \|\Phi\|=\|\varphi\|,\qquad \|\Phi^{-1}\|=\|\varphi^{-1}\|. \]

* For the definition of the spaces \(C,c,m\), see, for example, in \((^1)\). \(A\) is the space of analytic functions in the disk \(|z|<1\), continuous for \(|z|\le 1\), with norm
\[ \|f\|=\max_{|z|=1}|f(z)|. \]

We outline the proof of Theorem 1. Suppose first that \(\varphi\) is an isometry of \(P_1\) onto \(P_2\).* Consider the normed space of pairs \((x,y)\), \(x \in E_1,\ y \in E_2\), with pairs of the form \((x+z, y-\varphi(z))\), \(z \in P_1\), identified, and with norm

\[ \|(x,y)\|=\inf_{z\in P_1}\bigl(\|x+z\|+\|y-\varphi(z)\|\bigr). \]

The completion of this space may be taken as \(B\). If \(\varphi\) is an arbitrary isomorphism of \(P_1\) onto \(P_2\) and \(P_2=E_2\), then the proof of the theorem has in fact been given in \((^3)\) (pp. 211—212). The general case reduces to these two.

For what follows the following is essential.

Lemma 1. Each of the following sets is a metric compactum:

1) \(\mathfrak{N}(k,n)\), \(k,n\) natural numbers \((k\leq n)\), the set of all embedded pairs of Banach spaces \(\{P,R\}\), \(P\subset R\), \(\dim P=k\), \(\dim R=n\), with distance

\[ \rho(\{P_1,R_1\},\{P_2,R_2\})=\ln \inf_T \|T\|\cdot\|T^{-1}\|, \]

where \(T\) ranges over the isomorphisms of \(R_1\) onto \(R_2\) which map \(P_1\) onto \(P_2\) (in this case pairs \(\{P,R\}\), \(\{P'R'\}\) for which \(\rho(\{P,R\},\{P'R'\})=0\) are identified).

2) \(\mathfrak{G}(B)\) (\(B\) a finite-dimensional Banach space) — the set of all subspaces \(P\) in \(B\) with distance defined as the Hausdorff–Markus distance \(\theta(P_1,P_2)\) \((^4)\).

3) \(f(B_1,B_2,C)\) (\(B_1,B_2\) \(n\)-dimensional Banach spaces, \(C\geq 1\)) — the set of all isomorphisms \(\varphi\) of \(B_1\) onto \(B_2\) such that \(\max\{\|\varphi\|,\|\varphi^{-1}\|\}\leq C\), with distance

\[ \nu(\varphi_1,\varphi_2)=\max\{\|\varphi_1-\varphi_2\|,\|\varphi_1^{-1}-\varphi_2^{-1}\|\}. \]

For subspaces \(P\) and \(Q\) in \(E\) we shall consider the quantity

\[ (P,Q)=\inf_{x\in P,\ \|x\|=1}\rho(x,Q). \]

From a result of Auerbach and Taylor (see, for example, \((^1,^5)\)) the following follows.

Lemma 2. If \(P\) is an \(n\)-dimensional subspace in a Banach space \(E\), then there exists a direct complement \(Q\) to \(P\) in \(E\) such that

\[ \widehat{(P,Q)}\geq 1/n. \]

With the aid of Lemma 2 one establishes

Lemma 3. Let \(P,R\) and \(\widetilde P,\widetilde R\) be two pairs of Banach spaces; \(P\) a subspace in \(R\); \(\widetilde P\) a subspace in \(\widetilde R\); \(\dim P=\dim \widetilde P=n\); \(\Phi\) an isomorphism of \(R\) onto \(\widetilde R\); \(\varphi\) the restriction of \(\Phi\) to \(P\), mapping \(P\) onto \(\widetilde P\), with \(\max\{\|\varphi\|,\|\varphi^{-1}\|\}\leq C<\infty\). For the given \(\varepsilon>0\) there is a \(\delta=\delta(n,C,\varepsilon)\) such that, if for an isomorphism \(\varphi'\) of \(P\) onto \(\widetilde P\) the condition \(\nu(\varphi,\varphi')<\delta\) is fulfilled, then there exists an isomorphism \(\Phi'\) of \(R\) onto \(\widetilde R\), coinciding with \(\varphi'\) on \(P\) and such that \(\nu(\Phi,\Phi')<\varepsilon\).

It is easy to prove the following.

Lemma 4. Let \(E'\) be a subspace of a Banach space \(E\); \(Q\) a direct complement to a subspace \(P\) in \(E'\); \(\widehat{(P,Q)}=\beta>0\); \(R\) a subspace in \(E\) such that \(\tilde\theta(P,R)\leq \gamma<\beta\). Then \(Q\) is a direct complement to \(R\) in \(R+Q\) (\(R+Q\) is the closure of the set of elements of the form \(x+y\), \(x\in R,\ y\in Q\)), and

\[ \widehat{(R,Q)}\geq \beta-\gamma. \]

With the aid of Lemmas 2 and 4 one establishes

Lemma 5. Let \(P\) \((\dim P=n)\) be a subspace of \(a\)-universal \((a\geq 1)\) disposition in \(E\) with respect to \(\mathfrak{R}\). For the given \(\varepsilon>0\) there is a \(\delta=\delta(n,\varepsilon)>0\) such that, if for some subspace \(\widetilde P\) in \(E\) the condition \(\theta(P,\widetilde P)<\delta\) is fulfilled, then \(\widetilde P\) is a subspace of \((a+\varepsilon)\)-universal disposition in \(E\) with respect to \(\mathfrak{R}\).

Theorem 2. There exists a separable Banach space of almost universal disposition with respect to \(\mathfrak{R}\).

* The proof given in this case belongs to M. I. Kadets. It is simpler than the author’s original proof.

We outline the proof of Theorem 2. Consider sequences
\(\{\varepsilon_i\}_1^\infty\), \(\{a_i\}_1^\infty\), \(\varepsilon_i>0\), \(a_i>1\), \(\varepsilon_i\to 0\), \(a_i\to\infty\), and the sets

\[ \mathfrak{N}_i=\bigcup_{n=2}^{i+1}\bigcup_{k=1}^{n-1}\mathfrak{N}(k,n),\quad i=1,2,\ldots . \]

Let \(B_1\) be a finite-dimensional Banach space; let \(\{E_i^{(1)}\}_{i=1}^{m_1}\) be an \(\varepsilon_1\)-net in \(\mathfrak{S}(b_1)\); let \(\{P_i^{(1)}, R_i^{(1)}\}_{i=1}^{n_1}\) be an \(\varepsilon_1\)-net in \(\mathfrak{N}_1\) (\(P_i^{(1)}\) is a subspace of \(R_i^{(1)}\)); let \(\{\varphi_k^{(i,j)}\}_{k=1}^{r_1}\) be an \(\varepsilon\)-net in \(f(P_i^{(1)},E_j^{(1)},a_1)\), \(i=1,2,\ldots,n_1\), \(j=1,2,\ldots,m_1\) (which may also be empty). Applying Theorem 1 successively no more than \(n_1m_1r_1\) times, we obtain the existence of a finite-dimensional Banach space \(B_2\supset B_1\) with the following property: for the isomorphism \(\varphi_k^{(i,j)}\) there exists a subspace \(F_k^{(i,j)}\) in \(B_2\), \(F_k^{(i,j)}\supset E_j\), and an isomorphism \(\Phi_k^{(i,j)}\) of \(R_i^{(1)}\) onto \(F_k^{(i,j)}\) such that \(\Phi_k^{(i,j)}\) coincides with \(\varphi_k^{(i,j)}\) on \(P_i^{(1)}\) and

\[ \|\Phi_k^{(i,j)}\|=\|\varphi_k^{(i,j)}\|,\quad \|\Phi_k^{-1(i,j)}\|=\|\varphi_k^{-1(i,j)}\|, \]

\[ i=1,2,\ldots,n_1;\quad j=1,2,\ldots,m_1;\quad k=1,2,\ldots,r_1. \]

If, starting from \(\varepsilon_2,a_2,\mathfrak{N}_2,B_2\), we carry out the same reasoning, then the space \(B_3\supset B_2\) obtained thereby will again be finite-dimensional. Continuing this process without bound, we obtain a chain of embedded finite-dimensional spaces \(B_1\subset B_2\subset B_3\subset\cdots\), and we may assume that, for some separable Banach space \(B\), the \(B_i\) are subspaces of \(B\), \(i=1,2,\ldots\), and \(\bigcup_{i=1}^{\infty}B_i\) is dense in \(B\). With the aid of Lemmas 3 and 5 it is verified that \(B\) is the required space.

Theorem 3. A space of almost universal disposition with respect to \(\mathfrak{R}\) is almost universal with respect to \(\mathfrak{H}\). A space of universal disposition with respect to \(\mathfrak{R}\) is universal with respect to \(\mathfrak{H}\).

Theorem 4. Let \(t_0\) be the positive root of the equation \(t(1+t^2)^{1/2}=2\). For any topological space \(T\) and any \(a<\frac{4}{\sqrt2}\), \(C(T)^*\) is not a space of \(a\)-universal disposition with respect to \(\mathfrak{R}\).

Define the class \(\mathfrak{C}\) of infinite-dimensional Banach spaces by assuming that \(B\in\mathfrak{C}\) if, for every finite-dimensional subspace \(P\) in \(B\) and \(\varepsilon>0\), there exists a subspace \(E\subset B\), \(E\supset P\), \(\dim E=N=N(\varepsilon,P)\), and an isomorphism \(\varphi\) of \(E\) onto \(c^N\) such that \(\max\{\|\varphi\|,\|\varphi^{-1}\|\}<1+\varepsilon\). Thus, \(c,c_0,C,m\) are spaces of the class \(\mathfrak{C}\)*.

Theorem 5. In a separable space of the class \(\mathfrak{C}\) there exists a basis.

Corollary. In a separable space of almost universal disposition with respect to \(\mathfrak{R}\) there exists a basis.

Theorem 6. For separable Banach spaces of almost universal disposition with respect to \(\mathfrak{R}\) are almost isometric.

Definition 4. We shall call a Banach space \(E\) \(a\)-isotropic \((a\ge 1)\) if, for any finite-dimensional subspaces \(P,Q\), \(\dim P=\dim Q\), and any isomorphism \(\varphi\) of \(P\) onto \(Q\), there exists an isomorphism \(\Phi\) of \(E\) onto itself such that \(\Phi=\varphi\) on \(P\), \(\|\Phi\|\le a\|\varphi\|\), \(\|\Phi^{-1}\|\le a\|\varphi^{-1}\|\). If \(E\) is \(a\)-isotropic for every \(a>1\) (respectively for \(a=1\)), then we shall call \(E\) almost isotropic (respectively isotropic).

* Here \(C(T)\) is the space of all real bounded continuous functions on \(T\) with norm \(\|f\|=\sup_{x\in T}|f(x)|\).

** By \(c^n\) is denoted the space \(C(T)\), where \(T\) is a collection of \(n\) points.

*** It can be shown that for any metric compactum \(T\), \(C(T)\in\mathfrak{C}\).

Theorem 7. A separable Banach space of almost universal disposition with respect to \(\mathfrak{R}\) is almost isotropic.

Theorem 7 has some bearing on Mazur’s problem on rotations of Banach spaces ((\(^{1}\), p. 211)).

Theorem 8. There does not exist a separable Banach space of universal disposition with respect to \(\mathfrak{R}\).

I take this opportunity to express my sincere gratitude to M. I. Kadets for useful discussions.

Kharkov Automobile and Highway Institute

Received
18 I 1965

REFERENCES

\(^{1}\) S. Banach, Course of Functional Analysis, Kiev, 1948.
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\(^{3}\) A. Pełczyński, Studia Math., 19, 209 (1960).
\(^{4}\) I. Ts. Gokhberg, A. S. Markus, UMN, 14, no. 5, 135 (1959).
\(^{5}\) A. E. Taylor, Bull. Am. Math. Soc., 53, 614 (1947).
\(^{6}\) M. M. Day, Normed Linear Spaces, IL, 1959.