UDC 519.56

MATHEMATICS

V. I. AVERBUKH, O. G. SMOLYANOV

DIFFERENTIATION IN LINEAR TOPOLOGICAL SPACES

(Presented by Academician A. N. Kolmogorov on 30 IX 1966)

The present paper contains an exposition of the theory of differentiation in linear topological spaces (l.t.s.).* Definitions of the first and higher derivatives are introduced, and theorems are formulated on the mean value, on the differentiation of a composite and an inverse function, on the connection between total and partial derivatives, on interchanging the order of differentiation, on finding a primitive, Taylor’s formula is given, and the connection with the theory of variational derivatives is indicated.

A definition of the derivative of a mapping of one l.t.s. (not assumed to be normed) into another for various cases has been proposed by many authors \(\left({}^{1-6,\ 10,\ 11}\right)\). The definition adopted by us is a development of the definition of Sebastiao e Silva, who introduced, in essence (in our notation), \((X,\beta,\beta;\ldots;X,\beta,\beta)\)-derivatives.

Let \(X\) be a linear space; \(Y\) and \(H\) l.t.s., \(H \subseteq X\)**; \(\sigma\) some system of subsets of the space \(H\); \(\beta\) some system of bounded subsets of the space \(H\); \(\mathcal L(H,Y)\) the linear space of all linear continuous mappings of the space \(H\) into \(Y\); \(\mathcal L_{\beta}(H,Y)\) the l.t.s. obtained by endowing the space \(\mathcal L(H,Y)\) with the topology of uniform convergence on the sets of the system \(\beta\) \(\left({}^{7}\right)\).

Definition 1. A mapping \(f:X\to Y\) will be called \(\sigma\)-differentiable at a point \(x\in X\) with respect to the subspace \(H\) (briefly, \((H,\sigma)\)-differentiable at the point \(x\)) if there exists an element \(f'(x)\) of the space \(\mathcal L(H,Y)\) such that, for any set \(S\in\sigma\),
\[ \tau^{-1}(f(x+\tau h)-f(x))\to f'(x)\cdot h \]
as \(\tau\to 0\) (\(\tau\in R\)) uniformly with respect to \(h\) belonging to \(S\). The mapping \(f'(x)\) of the space \(H\) into \(Y\) will be called the \(\sigma\)-derivative of the mapping \(f\) at the point \(x\) with respect to the subspace \(H\) (briefly, the \((H,\sigma)\)-derivative at the point \(x\)). The mapping \(f':x\to f'(x)\) of the space \(X\) into \(\mathcal L_{\beta}(H,Y)\) will be called the \((\sigma,\beta)\)-derivative of the mapping \(f\) with respect to the subspace \(H\) (briefly, the \((H,\sigma,\beta)\)-derivative).

Definition 2. The \((\sigma_1,\beta_1;\ldots;\sigma_n,\beta_n)\)-derivative \(f^{(n)}\) with respect to the subspaces \(H_1,\ldots,H_n\) (briefly, the \((H_1,\sigma_1,\beta_1;\ldots;H_n,\sigma_n,\beta_n)\)-derivative) of a mapping \(f:X\to Y\) will be called the \((H_n,\sigma_n,\beta_n)\)-derivative of the \((H_1,\sigma_1,\beta_1;\ldots;H_{n-1},\sigma_{n-1},\beta_{n-1})\)-derivative \(f^{(n-1)}\). The mapping \(f\) will be called \((H_1,\sigma_1,\beta_1;\ldots;H_{n-1},\sigma_{n-1},\beta_{n-1};H_n,\sigma_n)\)-differentiable at the point \(x\) if the derivative \(f^{(n)}\) is defined at the point \(x\); the value of this derivative at the point \(x\) will be called the \((H_1,\sigma_1,\beta_1;\ldots;H_{n-1},\sigma_{n-1},\beta_{n-1};H_n,\sigma_n)\)-derivative at the point \(x\).

The derivative \(f^{(n)}(x)\) we can and shall also regard as a multilinear mapping of the product \(H_1\times\cdots\times H_n\) into \(Y\).

We shall call the \((H_1,\sigma_1,\beta_1;\ldots;H_n,\sigma_n,\beta_n)\)-derivative a strong (weak) \(H_1\ldots H_n\)-derivative if each of the systems \(\sigma_k,\beta_k\),

* Everywhere below, separable l.t.s. over the field of real numbers are considered.
** The symbol \(H \subseteq X\) means that \(H\) is embedded in \(X\) as a linear subspace.
*** Here and below we write \(f\cdot(x_1,\ldots,x_n)\) instead of \(f(x_1,\ldots,x_n)\), if \(f\) is a multilinear mapping of the product of linear spaces \(X_1\times\cdots\times X_n\) into \(Y\).

\(k=1,\ldots,n\), consists of all bounded (finite) subsets of the space \(H_k\). We shall simply write “derivative” instead of \(H_1\ldots H_n\)-derivative if \(H_1=\cdots=H_n=X\). For functions of a real variable \((f:R^1\to Y)\), weak and strong differentiability coincide.

Define by induction the difference of order \(n\) of a mapping \(f:X\to Y\) at the point \(x\in X\), for increments \(h_1,\ldots,h_n\in X\), by the equalities
\[ \Delta^n f(x;h_1,\ldots,h_n) = \Delta^{n-1}f(x+h_n;h_1,\ldots,h_{n-1}) - \Delta^{n-1}f(x;h_1,\ldots,h_{n-1}),\quad n\ge 2, \]
\[ \Delta^1 f(x;h_1)\equiv \Delta f(x;h_1)=f(x+h_1)-f(x). \]

Let \(A\) be a subset of a linear topological space \(E\). We shall denote by \(\overline{\Gamma A}\) the closed convex hull of \(A\).

Theorem 1 (on the mean). Let the space \(Y\) be locally convex. Then, if (for some \(\sigma_k,\beta_k,\ k=1,\ldots,n\)) the mapping \(f:X\to Y\) is \((H,\sigma_1,\beta_1;\ldots;H,\sigma_n)\)-differentiable at every point of the set
\[ P=\{x_0+\theta_1h_1+\cdots+\theta_nh_n,\ 0<\theta_i<1,\ i=1,\ldots,n\}, \]
\(x_0\in X,\ h_1,\ldots,h_n\in H\), then
\[ \Delta^n f(x_0;h_1,\ldots,h_n) \in \overline{\Gamma}\{f^{(n)}(x)\cdot(h_1,\ldots,h_n),\ x\in P\}. \]

For \(n=1\), Theorem 1 is a generalization of Lagrange’s theorem on finite increments.

The condition that the space \(Y\) be locally convex is essential for the validity of Theorem 1, as the following shows.

Example 1. Let \(Y\) be the space of all real measurable functions on the interval \([0,1]\), with the (linear, but not locally convex) topology of convergence in measure. For each \(\tau\in R^1\), denote by \(y_\tau\) the function on \([0,1]\) equal to 1 for \(t\le \tau\) and to zero for \(t>\tau\). The derivative of the mapping \(f:\tau\to y_\tau\) from the space \(R^1\) into \(Y\) is identically equal to zero; however, the mapping \(f\) is not constant.

Definition 3. A mapping \(f:X\to Y\) will be called continuous at the point \(x\in X\) with respect to the subspace \(H\) (in short, \(H\)-continuous at the point \(x\)) if the mapping \(h\to f(x+h)\) from the space \(H\) into \(Y\) is continuous at zero.

Theorem 2. If \(H\) is a metrizable l.t.s. and the mapping \(f:X\to Y\) is strongly \(H\)-differentiable at the point \(x\in X\), then it is \(H\)-continuous at this point.

The condition that the space \(H\) be metrizable is here in fact essential, as the following shows.

Example 2. Let, for each \(a>0\), \(K_a\) be the normed space of all real continuous finite functions defined on the real line whose supports are contained in the interval \((-a,a)\), with norm
\[ \|x\|_a=\sup_{t\in(-a,a)}|x(t)|; \]
let \(K\) be the (strict) inductive limit of the sequence of spaces \(K_n,\ n=1,2,\ldots\). Consider the countable set \(S\) of points \((x_{mk})\) of the space \(K\) ([8]):
\[ x_{mk}(t)=m^{-1}\varphi(t)+k^{-1}\varphi(t-m),\quad m,k=1,2,\ldots, \]
where \(\varphi\in K\). The characteristic function of the set \(S\) is strongly differentiable at the point \(x=0\), but is not continuous at this point.

For normed spaces, strong differentiability coincides with Fréchet differentiability.

Theorem 3 (on differentiating an inverse function). Let \(X\) be a metrizable l.t.s., and let \(f:X\to Y\) be a homeomorphism of an open set \(A\subset X\) onto an open set \(B\subset Y\); let \(g\) be the inverse homeomorphism. If the mapping \(f\) is strongly differentiable at the point \(x\in A\) and \(f'(x)\) is a linear homeomorphism of the space \(X\) onto \(Y\), then the mapping \(g\) is strongly differentiable at the point \(y=f(x)\) and
\[ g'(y)=(f'(x))^{-1}. \]

Let \(K,Z\) be l.t.s., \(K\subset Y\).

Theorem 4 (on differentiating a composite function). Let the mapping \(f:X\to Y\) be \((H,\sigma)\)-differentiable at the point \(x\in X\), let the mapping \(g:Y\to Z\) be strongly \(K\)-differentiable at the point \(f(x)\), and let the compatibility condition be satisfied:
\[ x_1-x_2\in H,\quad x_1,x_2\in X \Rightarrow f(x_1)-f(x_2)\in K. \]

Then, if in the space \(K\) there exists a countable fundamental system of bounded sets, the composition \((g\circ f): X\to Z^{-}(H,\sigma)\) is differentiable at the point \(x\) and
\[ (g\circ f)'(x)=g'(f(x))\circ f'(x). \]

Theorem 5 (on the relation between total and partial derivatives). Let the space \(Y\) be locally convex, and let the space \(H\subseteq X\) be the direct sum of two l.t.s. \(H=H_1\oplus H_2\). If for \(k=1,2\) the \((H_k,\sigma_k,\beta_k)\)-derivatives \(f_k\) of the mapping \(f:X\to Y\) are defined in some \(H\)-neighborhood* of the point \(x\in X\) and are \(H\)-continuous at this point, then the mapping \(f\) has at the point \(x\) an \((H,\beta)\)-derivative \(f'(x)\), where \(\beta\) is a system of bounded sets in \(H\) of the form \(B_1+B_2,\ B_1\in\beta_1,\ B_2\in\beta_2\), and
\[ f'(x)\cdot h=f_1(x)\cdot h_1+f_2(x)\cdot h_2, \]
where \(h=h_1+h_2,\ h_1\in H_1,\ h_2\in H_2\).

We formulate two theorems on the interchange of the order of differentiation.

Theorem 6 (Young’s theorem). Let the space \(Y\) be locally convex, and let the space \(H\subseteq X\) be the direct sum of two l.t.s. \(H=H_1\oplus H_2\). If both weak \(H_1\)- and \(H_2\)-derivatives of the mapping \(f:X\to Y\) are strongly \(H\)-differentiable at the point \(x\in X\), then at this point there exist a weak \(H_1H_2\)-derivative \(f_{12}(x)\) and a weak \(H_2H_1\)-derivative \(f_{21}(x)\), and
\[ f_{12}(x)\cdot(h_1,h_2)=f_{21}(x)\cdot(h_2,h_1),\qquad h_1\in H_1,\ h_2\in H_2. \]

Theorem 7 (Schwarz’s theorem). Let the space \(Y\) be locally convex, and let the space \(H\subseteq X\) be the direct sum of two l.t.s. \(H=H_1\oplus H_2\). If the mapping \(f:X\to Y\) is weakly \(H\)-differentiable in an \(H\)-neighborhood of the point \(x\in X\), and if the weak \(H_1H_2\)-derivative \(f_{12}\) is \(H\)-continuous at the point \(x\), then at this point there exists a weak \(H_2H_1\)-derivative \(f_{21}(x)\), and
\[ f_{12}(x)\cdot(h_1,h_2) = f_{21}(x)\cdot(h_2,h_1) = \lim_{(\tau_1,\tau_2)\to(0,0)} \tau_1^{-1}\tau_2^{-1}\Delta^2 f(x;\tau_1h_1,\tau_2h_2). \]

Theorem 8. Let the space \(Y\) be locally convex. If the \((H,\sigma,\beta)\)-derivative of the mapping \(f:X\to Y\) is \(H\)-continuous at the point \(x\in X\), then at this point the mapping \(f\) has an \((H,\beta)\)-derivative.

Theorem 9 (Taylor’s formula with remainder in Peano form). Let the space \(Y\) be locally convex. If the mapping \(f:X\to Y\) is \((\underbrace{H,\beta,\beta;\ldots,H,\beta,\beta}_{n})\)-differentiable at the point \(x\in X\), where \(\beta\) is some system of bounded sets in \(H\) containing all bounded sets of dimension \(n\), then
\[ f(x+h)=\sum_{k=0}^{n}\frac{1}{k!}f^{(k)}(x)\cdot (\underbrace{h,\ldots,h}_{k})+r(x,h,1), \]
where, for any set \(B\in\beta\),
\[ \tau^{-n}r(x,h,\tau)\to 0\quad\text{as }\tau\to 0 \]
uniformly with respect to \(h\in B\).

Theorem 10 (generalized Taylor formula with integral term). Let \(X\) be a complete l.t.s. possessing a countable fundamental system of bounded sets, \(Y\) a Fréchet space, and \(\alpha\) a mapping of the real line \(R^1\) into \(X\). If the mapping \(\alpha\) is \(n\) times differentiable at every point of the interval \([0,t]\) and the derivative \(\alpha^{(n)}\) is linear\(^9\) on \([0,t]\), and the mapping \(f\) is \(n\) times strongly differentiable in some neighborhood of the image of the segment \([0,t]\) under the mapping \(\alpha\) and the derivative \(f^{(n)}\) is continuous at every point of this neighborhood, then
\[ \begin{aligned} f(x_t)=f(x_0) &+\sum_{k=1}^{n}\frac{1}{k!}f^{(k)}(x_0)\cdot \sum_{p=k}^{n-1}\ \sum_{l_1+\cdots+l_k=p} t^p b(l_1,\ldots,l_k)\bigl(\alpha^{(l_1)}(0),\ldots\\ &\ldots,\alpha^{(l_k)}(0)\bigr) +\int_{0}^{t}\frac{(t-\tau)^{\,n-1}}{(n-1)!} \sum_{k=1}^{n} f^{(k)}(\alpha(\tau))\cdot \sum_{l_1+\cdots+l_k=n} c(l_1,\ldots,l_k)\cdot\\ &\qquad\qquad\qquad\qquad \cdot\bigl(\alpha^{(l_1)}(\tau),\ldots,\alpha^{(l_k)}(\tau)\bigr)\,d\tau . \end{aligned} \]

* We call \(H\)-neighborhoods of the point \(x\) sets of the form \(x+U\), where \(U\) is a neighborhood of zero in \(H\).

where \(x_t \equiv a(t)\),

\[ b(l_1,\ldots,l_k)=\frac{k!}{(l_1+\cdots+l_k)!}\,c(l_1,\ldots,l_k) = k!\left(\prod_{i=1}^k l_i!\prod_{j=1}^n m_j!\right)^{-1}, \]

\(m_j\) is the number of indices \(l_1,\ldots,l_k\) equal to \(j\).

Here the integral is understood in the Riemann sense.

Definition 3. Let \(\varphi\) be a mapping of the interval \([0,1]\) into the linear space \(X\); let \(\sigma\) be a system of subsets of the space \(X\). We shall call the mapping \(\varphi\) a \(\sigma\)-Lipschitz curve in \(X\) if there exists a set \(S\in\sigma\) such that \(\varphi(\tau_1)-\varphi(\tau_2)\in(\tau_1-\tau_2)S\) for all \(\tau_1,\tau_2\in[0,1]\).

Theorem 11 (on finding a primitive). Let \(X\) be an l.t.s., \(Y\) a sequentially complete locally convex space; let \(\sigma\) be some system of subsets of \(X\) containing all bounded sets of dimension two; let \(A\) be a connected open set in \(X\) such that every closed polygonal line (with a finite number of links) lying entirely in \(A\) is homologous to zero in \(A\); let \(\mathfrak A\) be some class of \(\beta\)-Lipschitz curves in \(A\), containing all polygonal lines lying entirely in \(A\), and let \(x_0\in A,\ y_0\in Y\). Then, if the mapping \(g:X\to\mathcal L_\beta(X,Y)\) is continuous and \(\sigma\)-differentiable at every point of the set \(A\), and the derivative \(g'(x)\) is symmetric* for all \(x\in A\), then there exists a mapping \(f:X\to Y\), \(\beta\)-differentiable at every point \(x\in A\), such that \(f'(x)=g(x)\) at every point \(x\in A,\ f(x_0)=y_0\).

This mapping is given by the formula

\[ f(x)=y_0+\int_0^1 g(\varphi(\tau))\,d\varphi(\tau), \]

where \(\varphi\) is any curve from the class \(\mathfrak A\) joining the points \(x_0\) and \(x\) \((\varphi(0)=x_0,\ \varphi(1)=x)\), and the integral is understood in the Riemann–Stieltjes sense.

Let us note in conclusion that the theory of the so-called variational derivatives can be constructed as a theory of derivatives of real-valued functionals defined on spaces of functions, with respect to subspaces of “basic functions”; in this case the variational derivatives (of first and, respectively, higher orders) turn out to be mappings from the original space of functions into the space of “generalized functions” (of one (\(^4\)) and, respectively, several variables).

The authors thank G. E. Shilov for his attention to their work, and also M. M. Vainberg and D. A. Raikov for a number of useful suggestions.

Moscow State University
named after M. V. Lomonosov

Received
1 IX 1966

REFERENCES

\(^1\) A. D. Michal, Bull. Am. Math. Soc., 45, 529 (1939).
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\(^3\) M. M. Vainberg, Ya. L. Entelson, Mat. sborn., 45, No. 4, 417 (1958).
\(^4\) M. D. Donsker, J. L. Lions, Acta Math., 108, 147 (1962).
\(^5\) J. Sebastião e Silva, a) Atti Acc. Lincei Rend., 20, fasc. 6, 743 (1956); b) 21, fasc. 1–2, 40 (1956).
\(^6\) J. Sebastião e Silva, Conceitos de função diferenciável em espaços localmente convexos, Publ. Centre de Estudos Mat. de Lisboa, 1956.
\(^7\) N. Bourbaki, Topological Vector Spaces, IL, 1959.
\(^8\) O. G. Smolyanov, Vestn. Mosk. univ., No. 1, 26 (1965).
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\(^10\) M. Balanzat, C. R., 251, 2459 (1960).
\(^11\) A. Bastiani, Differentiabilité dans les espaces localement convexes. Distructures, Thèse doct. sci. math., Univ. Paris, 1962.

* By the dimension of a set in a linear space we mean the dimension of the minimal linear manifold containing this set.

* We call an element \(a\) of the space \(\mathcal L\bigl(X,\mathcal L(X,Y)\bigr)\) symmetric* if

\[ (a\cdot h_1)\cdot h_2=(a\cdot h_2)\cdot h_1 \]

for all \(h_1,h_2\in X\).