Abstract
Full Text
UDC 513.88:513.83
MATHEMATICS
D. P. MILMAN
ON SUBLINEAR EXTENSIONS OF FUNCTIONALS
(Presented by Academician I. G. Petrovskii on 14 X 1968)
1. A sequence \(\{p_n\}_1^\infty\) of sublinear functionals defined on a vector subspace \(E_0\) of a Fréchet space \(E\) will be called consistent* (with the topology of \(E\)) if, for every convergent sequence \(\{x_n\}_1^\infty\) of elements of \(E_0\), \(\lim_{n\to\infty} x_n=x_0\), from the relations
\[ \lim_{m,n\to\infty} p_k(x_n-x_m)=0,\qquad k=1,2,\ldots, \]
it follows that
\[ x_0\in E_0,\quad \lim_{n\to\infty} p_k(x_n-x_0)=\lim_{m\to\infty} p_k(x_0-x_m)=0,\qquad k=1,2,\ldots . \]
We shall call a functional \(\widetilde p_k\) an extension of \(p_k\) from \(E_0\) to \(E\) if \(\widetilde p_k\) is defined, sublinear, and continuous on \(E\), and coincides with \(p_k\) on \(E_0\). We shall call this extension minimal if there exists no extension \(q_k\), different from \(\widetilde p_k\), for which \(q_k(x)\leq \widetilde p_k(x)\), \(x\in E\).
The result of this article is the following:
Theorem. Let \(\{p_k\}_1^\infty\) be a consistent sequence of sublinear functionals on a vector subspace \(E_0\) of a Fréchet space \(E\), and let \(E_0\) be of the second category in itself. Then each of the functionals \(p_k\) admits a minimal extension \(\widetilde p_k\) from \(E_0\) to \(E\). Moreover, if \(\psi_k(x)\) is a continuous seminorm on \(E\) such that \(|p_k(x)|\leq \psi_k(x)\) for all \(x\in E_0\), then \(\widetilde p_k\) can be chosen so that \(|\widetilde p_k(x)|\leq \psi_k\) for all \(x\in E\).***
2. Let us note that this theorem contains three fundamental principles of linear analysis. Let us verify this.
2a. Let \(E\) and \(E_1\) be Fréchet spaces; let \(\{Q_n\}_1^\infty\) be a sequence of seminorms defining the topology in \(E_1\); let \(\{A_\alpha\}_{\alpha\in S}\) be a collection of linear continuous operators mapping \(E\) into \(E_1\), and suppose, by assumption, that \(\{A_\alpha x\}_{\alpha\in S}\) is a bounded set for each \(x\in E\). The latter means that \(p_n(x)=\sup_{\alpha\in S} Q_n(A_\alpha x)<\infty\) for any \(x\in E\), \(n=1,2,\ldots\). It is verified directly that \(\{p_n\}_1^\infty\) is a consistent sequence of sublinear functionals defined on \(E_0=E\). From our theorem it follows directly that the functionals \(p_n\) are continuous on \(E\) (and hence bounded on bounded subsets of \(E\)). Hence, \(\lim_{x\to 0} A_\alpha x=0\) uniformly in \(\alpha\in S\).
Thus, the theorem contains the principle of uniform boundedness.
2b. Let \(E\) and \(E_1\) be Fréchet spaces; let \(\{Q_n\}_1^\infty\) be a sequence of seminorms defining the topology in \(E_1\); let \(E_0\) be a vector subspace dense in \(E\), of the second category in itself; let \(A:E_0\to E_1\) be a closed linear operator \((AE_0\subset E_1)\). Put \(p_n(x)=Q_n(Ax)\), \(x\in E_0\), \(n=\)
* A Fréchet space is a separable locally convex space \(E\) with a countable set of seminorms determining a metric in which \(E\) is complete.
** Note that the consistency property of the sequence of functionals \(\{p_k\}_1^\infty\), upon extension to \(\{\widetilde p_k\}_1^\infty\), is preserved by virtue of the continuity of the latter.
\(=1,2,\ldots\). It is clear that \(\{p_n\}_1^\infty\) is a compatible sequence of sublinear functionals defined on \(E_0\). By our theorem all the functionals \(p_n\) are continuous on \(E_0\). This means that the operator \(A\) is continuous on \(E_0\) (from the fact that \(E_0\) is dense in \(E\), it also follows that \(E=E_0\)).
We see that this theorem contains the theorem on the closed graph.
2b. Let \(p_0(x)\) be a discontinuous linear functional on a vector subspace \(E^0\) of a Fréchet space \(E\), and let \(E_0\) be the closure of \(E^0\) in \(E\). Since \(p_0\) admits an obvious discontinuous and linear extension to \(E_0\), we may assume that the functional \(p_0\) is defined on \(E_0\). Putting in our theorem \(p_k=p_0\), \(k=1,2,\ldots\), we see that it implies the existence for \(p_0\) of a minimal extension \(p\) from \(E_0\) to \(E\). Below it is verified that the minimality of the sublinear functional \(p\) implies its linearity. If, in addition, \(|p_0(x)|\leq \psi(x)\), \(x\in E_0\), where \(\psi\) is a continuous seminorm on \(E\), then, by our theorem, \(p\) can be chosen so that \(|p(x)|\leq \psi(x)\), \(x\in E\). This gives us the Hahn—Banach theorem (in analytic form—for Fréchet spaces).
We shall verify the linearity of \(p\) by contradiction. Suppose that there exist \(x_1,y_1\in E\) such that \(p(x_1+y_1)<p(x_1)+p(y_1)\). The linear span \(\widetilde E\) of the pair of elements \(x_1,y_1\) is at most two-dimensional. It is immediately seen that in \(\widetilde E\) there exists a linear functional \(f_1\) for which \(f_1(x)\leq p(x)\) for all \(x\in\widetilde E\). Form the functional \(p_*(x)=\inf_{y\in\widetilde E}[f_1(y)+p(x-y)]\), \(x\in E\). It is directly verified that the functional \(p_*\) is finite and sublinear on \(E\), and \(p_*(x)\leq p(x)\) for \(x\in E\), \(p_*(y)\leq f_1(y)\) for \(y\in\widetilde E\). Since for \(y\in\widetilde E\) we have \(p_*(y)\leq f_1(y)\leq -p_*(-y)\leq p_*(y)\), it follows that \(p_*(y)=f_1(y)\) for \(y\in\widetilde E\). For \(z\in E_0\) we have \(-p(-z)\leq -p_*(-z)\leq p_*(z)\leq p(z)\) and \(\pm p_0(z)=p_0(\pm z)=p(\pm z)\). Hence \(p_*\) is a sublinear extension from \(E_0\) to \(E\), and \(p_*(x)\leq p(x)\) for \(x\in E\). Since \(p_*=f_1\) on \(\widetilde E\), we have
\[
p_*(x_1)+p_*(y_1)=p_*(x_1+y_1)\leq p(x_1+y_1)<p(x_1)+p(y_1),
\]
whence either \(p_*(x_1)<p(x_1)\), or \(p_*(y_1)<p(y_1)\).
The contradiction obtained contradicts the minimality of the extension \(p\), which proves its linearity (cf. (1), pp. 1197—1198).
We have verified that the theorem formulated here contains the three basic principles of linear analysis. Its proof is given below. It relies only on Zorn’s theorem on ordered spaces and on Baire’s theorem on categories.
3. Proof of the theorem. a) Let \(\{P_n\}_1^\infty\) be a sequence of seminorms defining the topology in \(E\), and let \(\{p_n\}_1^\infty\) be the sequence of functionals indicated in the theorem. Form the functionals
\[
p_0(x)=\sum_{k=1}^{\infty}\frac{1}{2^k}\frac{|p_k(x)|}{1+|p_k(x)|},\quad x\in E_0;\qquad
P(x)=\sum_{n=1}^{\infty}\frac{1}{2^n}\frac{P_n(x)}{1+P_n(x)},\quad x\in E.
\]
The functional \(P\) may be chosen as a quasinorm in \(E\).* \(P\) defines a metric \(r(x,y)=P(x-y)\), determining the topology of \(E\), and, by assumption, \(E\) is complete in this metric. The functional \(p_0\) has in \(E_0\) all the properties of a quasinorm, except positivity on all elements of \(E_0\setminus\{0\}\). The compatibility of the sequence \(\{p_n\}_1^\infty\) can be expressed by the requirement: if \(\{x_n\}_1^\infty\subset E_0\), \(\lim_{n\to\infty}x_n=x_0\), and \(\lim_{n,m\to\infty}p_0(x_n-x_m)=0\), then
\[
x_0\in E_0\quad\text{and}\quad \lim_{n\to\infty}p_0(x_n-x_0)=0.
\]
Denote \(P_0(x)=\max[p_0(x),P(x)]\), \(x\in E_0\). \(P_0\) may be chosen as a quasinorm on \(E_0\); moreover \(P_0(x)\geq P(x)\) for \(x\in E_0\). From the
* A functional \(P\) is called a quasinorm in \(E\) if it is subadditive, \(P(0)=0\), \(P(-x)=P(x)\), and \(\lim_{\lambda\to0}P(\lambda x)=0\) for every \(x\in E\); \(\lim_{n\to\infty}P(x_n)=0\) implies \(\lim_{n\to\infty}P(\lambda x_n)=0\) for \(\{x_n\}_1^\infty\subset E\) and \(-\infty<\lambda<\infty\), \(P(x)>0\) for \(x\in E\setminus\{0\}\).
from the consistency of \(\{p_n\}_1^\infty\) it follows that \(E_0\) is complete in the metric \(\rho(x,y)=P_0(x-y)\).
Thus, \(E_0\) is considered by us in two topologies: in the \(P\)-topology it is of second category in itself, while in the \(P_0\)-topology it is complete; moreover, \(P(x)\leq P_0(x)\) for \(x\in E_0\). Applying under these conditions, for example, the arguments from \((^2)\), pp. 64–65, where only Baire’s category theorem is used, we see that for every \(\varepsilon>0\) one can indicate a \(\delta(\varepsilon)>0\), \(\delta(\varepsilon)\leq\varepsilon\), such that \(P(x)\leq\delta(\varepsilon)\) entails \(P_0(x)\leq\varepsilon\) \((x\in E_0)\). A fortiori,
\[ P(x)\leq \delta(\varepsilon)\quad \text{entails}\quad p_0(x)\leq \varepsilon . \tag{1} \]
For a natural number \(k\), consider \(\varepsilon=1/2^{k+1}\) and denote \(\delta_k=\delta(\varepsilon)\). Choose a natural number \(N_k\) so that \(1/2^{N_k}<\delta_k\), and denote
\[
\eta_k=(\delta_k-2^{-N_k})/(1-\delta_k).
\]
In addition, for what follows denote
\[ \chi_k(x)=\frac{1}{\eta_k}\max_{1\leq m\leq N_k} P_m(x)\quad (x\in E). \tag{2} \]
Let us note that from \(\chi_k(x)\leq 1,\ x\in E_0\), it follows that
\[ \sum_{m=1}^{N_k}\frac{1}{2^m}\frac{P_m(x)}{1+P_m(x)} \leq \frac{\eta_k}{1+\eta_k}\left(1-\frac{1}{2^{N_k}}\right). \]
Since, moreover,
\[ \sum_{m=N_k+1}^{\infty}\frac{1}{2^m}\frac{P_m(x)}{1+P_m(x)} \leq \frac{1}{2^{N_k}}, \]
we have
\[ P(x)\leq \frac{\eta_k}{1+\eta_k}\left(1-\frac{1}{2^{N_k}}\right) +\frac{1}{2^{N_k}} = \frac{\eta_k+1/2^{N_k}}{1+\eta_k} =\delta_k=\delta(\varepsilon). \]
Therefore, from \(\chi_k(x)\leq 1,\ x\in E_0\), it follows that \(p_0(x)\leq\varepsilon\), and this, by virtue of the definition of \(p_0(x)\), gives
\[
|p_k(x)|/(1+|p_k(x)|)\leq 2^k\varepsilon.
\]
Since we chose \(\varepsilon=1/2^{k+1}\), we obtain \(|p_k(x)|\leq 1\).
Thus, \(\chi_k(x)\leq 1,\ x\in E_0\), entails \(|p_k(x)|\leq 1\). From the positive homogeneity of the functionals \(\chi_k\) and \(p_k\) it follows that
\[ |p_k(x)|\leq \chi_k(x)\quad \text{for } x\in E_0. \tag{3} \]
Formula (3) shows that \(p_k\) is continuous on \(E_0,\ k=1,2,\ldots\).
b) Let \(\psi_k(x)\) be some symmetric and continuous sublinear functional on \(E\) satisfying \(|p_k(x)|\leq\psi_k(x)\) for \(x\in E_0\), for example, \(\psi_k=\chi_k\). It is not hard to verify that the functional
\[ q_k(x)=\inf_{y\in x-E_0}\,[p_k(x-y)+\psi_k(y)],\quad x\in E, \tag{4} \]
is sublinear (this is the enveloping sublinear functional for \(p_k\) and \(\psi_k\))* . Moreover, \(q_k(x)\leq\psi_k(x)\) for \(x\in E\) and \(q_k(y)\leq p_k(y)\) for \(y\in E_0\); the first of these inequalities entails the continuity of \(q_k\) in \(E\). In addition, for any \(\varepsilon>0\) one can indicate such a \(y_\varepsilon\in x-E_0\) that, for the given \(x\in E\),
\[
p_k(x-y_\varepsilon)+\psi_k(y_\varepsilon)<q_k(x)+\varepsilon.
\]
Hence, for \(x\in E_0\),
\[
p_k(x)\leq \psi_k(y_\varepsilon)-p_k(y_\varepsilon)+p_k(x)
\leq p_k(x-y_\varepsilon)+\psi_k(y_\varepsilon)<q_k(x)+\varepsilon.
\]
In view of the arbitrariness of \(\varepsilon>0\), for the given \(x\in E_0\) we obtain \(p_k(x)\leq q_k(x)\). The inequality in the other direction was noted above. Thus,
\[
q_k(x)=p_k(x)\quad \text{for } x\in E_0.
\]
Thus, we have proved the existence of a (continuous and sublinear) extension \(q_k\) of the functional \(p_k\) from \(E_0\) to \(E\). Moreover, it has been proved that for every symmetric and continuous sublinear functional \(\psi_k(x)\) in \(E\) satisfying \(|p_k(x)|\leq\psi_k(x),\ x\in E_0\),
* See \((^1)\), pp. 1199–1200, where verification of sublinearity is given.
$q_k(x)$ can be chosen so that $q_k(x) \leq \psi_k(x)$ for $x \in E$; since
$-q_k(x) \leq q_k(-x) \leq \psi_k(-x)=\psi_k(x)$, it follows that
$|q_k(x)| \leq \psi_k(x)$ for $x \in E$.
c) Let $K$ denote the set of all extensions $\tilde p_k$ of the functional $p_k$ from $E_0$ to $E$ for which $\tilde p_k(x) \leq q_k(x)$ for $x \in E$; obviously, $q_k \in K$.
Order $K$ as follows: if $r_1, r_2 \in K$ and $r_1(x) \geq r_2(x)$ for all $x \in E$, then we shall write $r_1 \prec r_2$, where $\prec$ is the symbol of the order relation.
We shall prove that the ordered space $K$ is inductive, i.e., every linearly ordered subset $\{r_t\}_{t<\vartheta}$ of the space $K$ has a majorant in $K$. By Zorn’s theorem it will then follow that there exists a maximal element in $K$; this element will be the required minimal extension $\tilde p_k$ of the functional $p_k$ from $E_0$ to $E$. Moreover,
$\tilde p_k(x) \leq q_k(x) \leq \psi_k(x)$, and therefore $|\tilde p_k(x)| \leq \psi_k(x)$ for $x \in E$.
Let $t_1 \prec t_2 \prec \vartheta$. From the subadditivity of $r_{t_1}$ and $r_{t_2}$ it follows that
$-r_{t_1}(-x) \leq -r_{t_2}(-x) \leq r_{t_2}(x) \leq r_{t_1}(x)$ for $x \in E$.
Hence it follows that the functional
$r_{\vartheta}(x)=\inf_{t<\vartheta} r_t(x)$, $x \in E$, is finite, and also
$r_{\vartheta}(x) \leq r_t(x)$ for $t<\vartheta$ and $x \in E$. If $x,y \in E$, then
$r_{\vartheta}(x+y) \leq r_t(x+y) \leq r_t(x)+r_t(y)$ for $t<\vartheta$. Therefore the functional $r_{\vartheta}$ is finite and subadditive; its positive homogeneity is obvious. Thus $r_{\vartheta}\in K$; moreover, $r_{\vartheta}$ is a majorant of $\{r_t\}_{t<\vartheta}$ in the sense of the order $\prec$.
This proves that $p_k$ has a minimal extension $\tilde p_k$ from $E_0$ to $E$ (cf. (1), p. 1198), and in the case $|p_k(x)| \leq \psi_k(x)$ for $x \in E_0$ one can choose $\tilde p_k$ so that $|\tilde p_k(x)| \leq \psi_k(x)$ for $x \in E$.
Odessa Electrotechnical Institute of Communications
named after A. S. Popov
Received
9 X 1968
REFERENCES
- D. P. Milman, Izv. Akad. Nauk SSSR, Ser. Mat., 27, 6, 1189 (1963).
- M. M. Day, Normed Linear Spaces, IL, 1961.