Let ■ be the linear space of all C<sup>1</sup> vector fields X on a compact n-dimensionalC<sup>∞</sup> Riemann manifold（n≥2）,endowed with the C<sup>1</sup> norm ‖X‖<sub>...Let ■ be the linear space of all C<sup>1</sup> vector fields X on a compact n-dimensionalC<sup>∞</sup> Riemann manifold（n≥2）,endowed with the C<sup>1</sup> norm ‖X‖<sub>1</sub>.Write θ（X）for the numberof contractible periodic orbits of X∈（?）,which may be finite or infinite.Let （?）<sup>*</sup> be the set ofall X∈（?） possessing the property that X has a neighbourhood （?） such that every Y∈（?） hasonly a finite number of singularities and at most a countable number of periodic orbits.Inthis paper,it is shown that any given S∈（?） has a neighbourhood （?） in （?） together with anumber λ=λ（?）】0 such that θ（X）≤λfor all X∈（?）.展开更多
The goal of this paper is to confirm that the unitary group U(H) on an infinite dimensional complex Hilbert space is a topological group in its strong topology, and to emphasize the importance of this property for app...The goal of this paper is to confirm that the unitary group U(H) on an infinite dimensional complex Hilbert space is a topological group in its strong topology, and to emphasize the importance of this property for applications in topology. In addition, it is shown that U(H) in its strong topology is metrizable and contractible if H is separable. As an application Hilbert bundles are classified by homotopy.展开更多
SINCE 1956, Michael’s continuous selection theory has been applied to functional analysis,topology, approximation theory and other mathematical fields. In this letter, the concept ofthe pseudo-lower semicontinuity is...SINCE 1956, Michael’s continuous selection theory has been applied to functional analysis,topology, approximation theory and other mathematical fields. In this letter, the concept ofthe pseudo-lower semicontinuity is introduced, and a convex structure of metric space is de-fined. A continuous selection theorem for pseudo-lower semicontinuity is given. This展开更多
基金Supported by National Natural Science Foundation of China
文摘Let ■ be the linear space of all C<sup>1</sup> vector fields X on a compact n-dimensionalC<sup>∞</sup> Riemann manifold（n≥2）,endowed with the C<sup>1</sup> norm ‖X‖<sub>1</sub>.Write θ（X）for the numberof contractible periodic orbits of X∈（?）,which may be finite or infinite.Let （?）<sup>*</sup> be the set ofall X∈（?） possessing the property that X has a neighbourhood （?） such that every Y∈（?） hasonly a finite number of singularities and at most a countable number of periodic orbits.Inthis paper,it is shown that any given S∈（?） has a neighbourhood （?） in （?） together with anumber λ=λ（?）】0 such that θ（X）≤λfor all X∈（?）.
文摘The goal of this paper is to confirm that the unitary group U(H) on an infinite dimensional complex Hilbert space is a topological group in its strong topology, and to emphasize the importance of this property for applications in topology. In addition, it is shown that U(H) in its strong topology is metrizable and contractible if H is separable. As an application Hilbert bundles are classified by homotopy.
文摘SINCE 1956, Michael’s continuous selection theory has been applied to functional analysis,topology, approximation theory and other mathematical fields. In this letter, the concept ofthe pseudo-lower semicontinuity is introduced, and a convex structure of metric space is de-fined. A continuous selection theorem for pseudo-lower semicontinuity is given. This
