V. I. Gurarii and M. I. Kadets

On Minimal Systems and Quasi-Complements in Banach Space

(Presented by Academician A. N. Kolmogorov, 23 II 1962)

Let \(E\) be a separable Banach space, and let \(E^*\) be its conjugate. A collection \(\{x_i\}_1^\infty\) of elements of the space \(E\) is called minimal if each \(x_i\) does not belong to the closure of the linear span of the remaining elements. A collection of linear functionals \(\{f_i\}_1^\infty\) will be called conjugate to \(\{x_i\}_1^\infty\) if

\[ f_i(x_j)=\delta_{ij}\quad (x_i\in E,\ f_i\in E^*;\ i,\ j=1,\ 2,\ldots). \]

Every minimal system has a conjugate one. A set \(G\subset E^*\) is called total with respect to a subspace \(E_1\subset E\) if, for any element \(x\in E_1\), from the condition \(g(x)=0\) for all \(g\in G\) it follows that \(x=\theta\). We shall agree to call a subspace \(E_1\subset E\) nontrivial if both it itself and the quotient space \(E/E_1\) are infinite-dimensional; in what follows only nontrivial subspaces are considered.

Theorem 1. Suppose that in a subspace \(E_1\subset E\) a complete minimal system \(\{x_i\}_1^\infty\) is defined, and that its conjugate system \(\{f_i\}_1^\infty\) \((f_i\in E^*)\) is total with respect to \(E_1\). Then the system \(\{x_i\}_1^\infty\) can be extended to a minimal system, complete in \(E\) and having a total (with respect to \(E\)) conjugate system.

Proof. In the quotient space \(E/E_1\) (as in every separable Banach space \((^1)\)) there exists a complete minimal system \(\{Y_i\}_1^\infty\) with a total conjugate system \(\{\Phi_i\}_1^\infty\). From each adjacent class \(Y_i\) choose one element \(y_i\in E\), and to each linear functional \(\Phi_i\) assign the linear functional \(\varphi_i\) generated by it, defined in \(E\):

\[ \varphi_i(x)=\Phi_i(x+E_1)\quad (x\in E). \]

It is easy to verify that the set \(\{x_i\}_1^\infty\cup\{y_i\}_1^\infty\) is complete in \(E\), and the set \(\{f_i\}_1^\infty\cup\{\varphi_i\}_1^\infty\) is total. With the aid of the equalities

\[ z_n=y_n+\sum_{i=1}^{n}\lambda_{ni}x_i;\qquad \psi_n=f_n+\sum_{i=1}^{n}\mu_{ni}\varphi_i \quad (n=1,\ 2,\ldots), \]

where

\[ \lambda_{ni}=-f_i(y_n);\quad \mu_{ni}=-f_n(y_i);\quad \lambda_{nn}+\mu_{nn}=1-f_n(y_n), \]

we define new systems \(\{z_i\}_1^\infty\) and \(\{\psi_i\}_1^\infty\). A direct calculation shows that the set \(\{x_i\}_1^\infty\cup\{z_i\}_1^\infty\) is a complete minimal system in \(E\) with total conjugate system \(\{\psi_i\}_1^\infty\cup\{\varphi_i\}_1^\infty\).

Theorem 2. Let \(\{x_i\}_1^\infty\cup\{y_i\}_1^\infty\) be a minimal system in \(E\), and suppose that \(\{x_i\}_1^\infty\) is complete in the subspace \(P\subset E\). Form a new minimal sequence of elements

\[ u_i=\alpha_i x_i+y_{k_i}\quad (i=1,\ 2,\ldots), \]

where \(\alpha_i\) are arbitrary numbers, and \(\{y_{k_i}\}_{i=1}^{\infty}\) is some subset of the system \(\{y_i\}_{1}^{\infty}\). Denote by \(R\) the closure of the linear span of the system \(\{u_i\}_{i=1}^{\infty}\). If the system conjugate to \(\{u_i\}_{1}^{\infty}\) is total with respect to \(R\), then \(R \cap P = \theta\).

Proof. Let \(x\) be an arbitrary element of the closure of the linear span of the system \(\{x_i\}_{1}^{\infty} \cup \{y_i\}_{1}^{\infty}\):

\[ x=\lim_{n\to\infty}\left\{\sum_{i=1}^{n} a_{ni}x_i+\sum_{i=1}^{n} b_{ni}y_i\right\}. \tag{1} \]

Since this system is minimal, the limits exist

\[ \lim_{n\to\infty} a_{ni}=a_i(x);\qquad \lim_{n\to\infty} b_{ni}=b_i(x)\quad (i=1,2,\ldots). \tag{2} \]

Suppose now that \(x\in P\cup R\). From the fact that \(x\in P\), it follows that \(b_i(x)=0\) \((i=1,2,\ldots)\), and from the fact that \(x\in R\), it follows that

\[ x=\lim_{n\to\infty}\sum_{i=1}^{n} c_{ni}u_i =\lim_{n\to\infty}\left\{\sum_{i=1}^{n} c_{ni}\alpha_i x_i+\sum_{i=1}^{n} c_{ni}y_{k_i}\right\}. \tag{3} \]

Comparing (1), (2), and (3), we see that

\[ \lim_{n\to\infty} c_{ni}=b_i=0\quad (i=1,2,\ldots); \]

since the system \(\{u_i\}_{1}^{\infty}\) has a total conjugate, the last equality implies \(x=\theta\).

Remark 1. The system \(\{u_i\}_{1}^{\infty}\) has a total conjugate, for example, in the case when the system \(\{x_i\}_{1}^{\infty}\cup\{y_i\}_{1}^{\infty}\) has this property.

Remark 2. Theorems 1 and 2 complement the results of V. G. Vinokurov \((^{2,3})\) and generalize some of them.

We shall call Banach spaces \(E_1\) and \(E_2\) \(\varepsilon\)-isometric if there exists an isomorphism \(T\) from \(E_1\) onto \(E_2\) such that for every \(x\in E_1\):

\[ (1-\varepsilon)\|x\|\leq \|Tx\|\leq (1+\varepsilon)\|x\|\quad (0<\varepsilon<1). \]

Theorem 3. Let \(\{x_i\}_{1}^{\infty}\) be a complete minimal system in a subspace \(P\subset E\). For a given \(\varepsilon>0\) there exists a sequence of positive numbers \(\varepsilon_i\) such that, for any sequence \(\{y_i\}_{1}^{\infty}\), \(y_i\in E\), satisfying the condition

\[ \|x_i-y_i\|<\varepsilon_i\quad (i=1,2,\ldots), \]

the linear mapping \(T\) taking \(x_i\) to \(y_i\) \((i=1,2,\ldots)\) will be an \(\varepsilon\)-isometry of the subspace \(P\) onto the closure of the linear span of \(\{y_i\}_{1}^{\infty}\).

Proof. Put \(\varepsilon_i=\varepsilon r_i/2^i\), where \(r_i\) is the distance of the element \(x_i\) from the linear span of the remaining elements of the system \(\{x_i\}_{1}^{\infty}\). Define on the set of linear combinations of the elements of this system a linear operator \(T\), putting

\[ Tx_i=y_i\quad (i=1,2,\ldots). \]

From the inequality

\[ \|x\|=\left\|\sum_{i=1}^{n}\alpha_i x_i\right\|\geq |\alpha_i|\,r_i\quad (i=1,2,\ldots) \]

it follows that

\[ |\alpha_i|\leq \frac{\|x\|}{r_i}. \]

Let us now estimate \(\|Tx\|\):

\[ \|Tx\|=\left\|\sum_{i=1}^{n}\alpha_i y_i+\sum_{i=1}^{n}\alpha_i(x_i-y_i)\right\|; \]

since

\[ \left\|\sum_{i=1}^{n}\alpha_i(x_i-y_i)\right\| \leq \sum_{i=1}^{n}\frac{\|x\|}{r_i}\cdot\frac{\varepsilon r_i}{2^i} \leq \varepsilon\|x\|, \]

it follows that

\[ (1-\varepsilon)\|x\|\leq \|Tx\|\leq (1+\varepsilon)\|x\|. \]

Extending the operator \(T\) by continuity to all of \(P\), we obtain the required \(\varepsilon\)-isometric mapping.

Let \(P\) and \(Q\) be subspaces in \(E\), with \(P\cap Q=\theta\). We shall call the sum of \(P\) and \(Q\) a quasi-direct sum and denote by \(P\dot{+}Q\) the closure of the set of elements of the form \(x+y\) \((x\in P,\ y\in Q)\). If \(P\dot{+}Q=E\), then each of the subspaces \(P\) and \(Q\) is called a quasicomplement of the other in \(E\).

Theorem 4. If \(P\) and \(Q\) are nontrivial subspaces in \(E\) and \(P\supset Q\), then for any \(\varepsilon>0\):

1) There exists a subspace \(\widetilde Q\subset E\), \(\varepsilon\)-isometric to \(Q\) and quasicomplementary to \(P\).

2) There exists a subspace \(\widetilde P\subset E\), \(\varepsilon\)-isometric to \(P\) and quasicomplementary to \(Q\).

Proof. By Theorem 1, in \(E\) there exists a complete minimal system
\(\{x_i\}_1^\infty\cup\{y_i\}_1^\infty\cup\{z_i\}_1^\infty\), with conjugate system total relative to \(E\), such that \(\{x_i\}_1^\infty\) is a complete system in \(Q\), and \(\{x_i\}_1^\infty\cup\{y_i\}_1^\infty\) is complete in \(P\). By Theorem 3 one can choose a sequence \(\{\varepsilon_i\}_1^\infty\), \(\varepsilon_i>0\), so that the closure \(\widetilde Q\) of the linear span of the elements \(u_i=x_i+\varepsilon_i z_i\) is \(\varepsilon\)-isometric to \(Q\). Since the system \(\{u_i\}_1^\infty\) has, evidently, a conjugate system total (relative to \(\widetilde Q\)), by Theorem 2 \(\widetilde Q\cap P=\theta\). Moreover \(P\dot{+}\widetilde Q\) contains all elements of the set
\(\{x_i\}_1^\infty\cup\{y_i\}_1^\infty\cup\{z_i\}_1^\infty\), and therefore \(P\dot{+}\widetilde Q=E\), which proves assertion 1). Assertion 2) is proved similarly.

The theorem proved generalizes Mackey’s result \({}^{(4)}\) on the existence of a quasicomplement to any subspace in a separable Banach space.

Corollary 1. If \(P\) is a nontrivial subspace in \(E\), then for any \(\varepsilon>0\), \(E=P\dot{+}\widetilde P\), where \(\widetilde P\) is \(\varepsilon\)-isometric to \(P\).

Since in every Banach space there exists an infinite-dimensional subspace with a basis, it follows directly from Theorem 4 that

Theorem 5. In a separable Banach space every infinite-dimensional subspace has a quasicomplement with a basis.

Corollary 2. A separable Banach space \(E\) can be represented as the quasi-direct sum of its subspaces with bases.

Kharkov Automobile and Highway Institute
Kharkov Higher Military Aviation School

Received
23 II 1962

REFERENCES

\({}^{1}\) A. I. Markushevich, DAN, 41, 241 (1943).
\({}^{2}\) V. G. Vinokurov, DAN, 81, 337 (1951).
\({}^{3}\) V. G. Vinokurov, DAN, 85, 685 (1952).
\({}^{4}\) G. W. Mackey, Bull. Am. Math. Soc., 52, 322 (1946).