Bernstein’s theorem for polynomial maps of complex topological vector spaces. (English) Zbl 0581.46040
Let X be a complex Hausdorff topological vector space and let Y be a complex Hausdorff locally convex and quasi-complete topological vector space. Let P stand for a generalized polynomial map from X to Y of the form \(P=P_ 0+P_ 1+...+P_ N\), \(P_ N\neq 0\), where \(P_ m: X\to Y\), \(m=1,2,...,N\), are m-homogeneous generalized polynomial maps and \(P_ 0: X\to Y\) is a constant map and let P’(x,x) be the directional derivative of P at x in the direction of x.
Main result: Let \(\Omega\) and \(\Gamma\) be closed balanced convex sets in X and Y, respectively, and let p and q be their gauges. If \(P: X\to Y\) is a generalized polynomial map of degree N and \(P: \Omega\) \(\to \Gamma\), then, for each \(x\in \Omega\), the inequality \(q\circ P'(x,y)\leq Np(x)\) holds. The result is best possible and equality holds for an N- homogeneous generalized polynomial map P and \(x\in \Omega\), such that \(q\circ P(x)=p(x)=1\). (In case \(X=Y={\mathbb{C}}\) this result is due to Bernstein.)
Main result: Let \(\Omega\) and \(\Gamma\) be closed balanced convex sets in X and Y, respectively, and let p and q be their gauges. If \(P: X\to Y\) is a generalized polynomial map of degree N and \(P: \Omega\) \(\to \Gamma\), then, for each \(x\in \Omega\), the inequality \(q\circ P'(x,y)\leq Np(x)\) holds. The result is best possible and equality holds for an N- homogeneous generalized polynomial map P and \(x\in \Omega\), such that \(q\circ P(x)=p(x)=1\). (In case \(X=Y={\mathbb{C}}\) this result is due to Bernstein.)
MSC:
46G20 | Infinite-dimensional holomorphy |
Keywords:
closed balanced convex sets; Gâteaux-analytic maps; complex Hausdorff locally convex and quasi-complete topological vector space; generalized polynomial; directional derivativeReferences:
[1] | Tung, Proc. Amer. Math. Soc. 85 pp 73– (1982) |
[2] | Tung, Proc. Amer. Math. Soc. 83 pp 103– (1981) |
[3] | Hervé, Lecture Notes in Mathematics 198 (1971) |
[4] | DOI: 10.1112/jlms/s2-1.1.57 · Zbl 0179.37901 · doi:10.1112/jlms/s2-1.1.57 |
[5] | Hervé, Lecture Notes in Mathematics 364 (1974) |
[6] | DOI: 10.5802/aif.332 · Zbl 0176.09903 · doi:10.5802/aif.332 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.