Sufficient conditions for the differentiability of mappings of locally convex spaces (Q1089584)

Let E,F be locally convex spaces, U an open subset of E, \(\sigma(E)\) (resp. \(b(E)\)) the system of all nonempty finite (resp. bounded) subsets of \(E\) and \({\mathcal A}^ a \) system of subsets of \(E\) such that \(\sigma(E)\subset {\mathcal A}\subset b(E)\), \({\mathbb{R}}{\mathcal A}\subset {\mathcal A}\) and for any \(A_ 1,A_ 2\in {\mathcal A}\) there is \(A_ 3\in {\mathcal A}\) such that \(A_ 1\cup A_ 2\subset A_ 3\), \(A_ 1+A_ 2\subset A_ 3\). We say that 1) \(x_ n\to^{{\mathcal A}}x_ 0\) if there is a sequence \((t_ n)\) in \(R\) such that \(t_ n\to 0\) and \(((x_ n-x_ 0)/t_ n)\subset A\) for some \(A\in {\mathcal A};\) 2) \(f:U\to F\) is \({\mathcal A}\)-continuous if \((x_ n)_ 0^{\infty}\subset U\), \(x_ n\to^{{\mathcal A}}x_ 0\) implies \(f(x_ n)\to f(x_ 0);\) 3) \(E\) is \({\mathcal A}\)-countably compact if each sequence in any \(A\in {\mathcal A}\) contains an \({\mathcal A}\)-convergent subsequence; 4) \(f\) is \({\mathcal A}\)-differentiable at \(x_ 0\in U\) if there is \(f'(x_ 0)\in L_{{\mathcal A}}(E,F)\) (=the set of all linear \({\mathcal A}\)-continuous mappings from \(E\) to \(F\) endowed with the topology of uniform convergence on sets from \({\mathcal A})\) such that \(\lim_{t\to 0}r(th)/t=0\) uniformly on each \(A\in {\mathcal A}\), where \(r(k)=f(x_ 0+k)-f(x_ 0)-f'(x_ 0)k.\) Set \(L^ 1_{{\mathcal A}}(E,F)=L_{{\mathcal A}}(E,F)\), \(D^ 1_{{\mathcal A}}(U,F)=\{f:U\to F:\) f is \({\mathcal A}\)-differentiable at each \(x\in U\}\), \(f^{(1)}:U\to L_{{\mathcal A}}(E,F)(f^{(1)}(x)=f'(x))\), \(L^ m_{{\mathcal A}}(E,F)=L_{{\mathcal A}}(E,L_{{\mathcal A}}^{m-1}(E,F))\), \(D^ m_{{\mathcal A}}(U,F)=\{f\in D_{{\mathcal A}}^{m-1}(U,F):f^{(m-1)}\in D_{{\mathcal A}}(U,L_{{\mathcal A}}^{m-1}(E,F))\}\), \(f^{(m)}:U\to L^ m_{{\mathcal A}}(E,F)\) \((f^{(m)}(x)=(f^{(m-1)})'(x))\), \(C^ 0_{{\mathcal A}}(U,F)=\{f:U\to F: f\) is \({\mathcal A}\)-continuous\(\}\), \(C^ m_{{\mathcal A}}(U,F)=\{f\in D^ m_{{\mathcal A}}(U,F):f^{(m)}\in C^ 0_{{\mathcal A}}(U,L^ m_{{\mathcal A}}(E,F))\}.\) Theorem 1. Let \(n\geq 0\), \(f\in C^ 0_{{\mathcal A}}(U,F)\) be such that for each \(x\in U\) and each \(k\), \(0\leq k\leq n\), there is a \(k\)-linear mapping \(\phi_ k(x):E^ k\to F\) such that \(r_ n(x,0)=0\) and, for each \(A_ 1,A_ 2\in {\mathcal A}\), \(\lim_{t\to 0}r_ n(x+tu,th)/t^ n=0\) uniformly for \(u\in A_ 1\) and \(h\in A_ 2\), where \(r_ n\) is defined by \(f(x+h)=\sum^{n}_{0}\phi_ k(x)h^ k/k!+r_ n(x,h)\). Then \(f\in C^ m_{{\mathcal A}}(U,F)\) and \(f^{(k)}(x)\) is the symmetrization of the multilinear map \(\phi_ k(x)\).