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2004•Journal of Shaoyang UniversityRequires access

A Comparison between Topologies of Two Classes of Convergence in B(I) Space

Jingwen Li

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Abstract

Pointwise convergence has no completeness in bounded function space B(I). The topology of uniform convergence has three characteristic properties in B(I). A complete norm topology stronger than the topology of poingwise convergence can only be one of uniform convergences. Arzela gave a necessary and sufficient condition of continuation of limit functions in sequences of continuous functions, among which topology can not be a norm topology.

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Pointwise convergence has no completeness in bounded function space B(I). The topology of uniform convergence has three characteristic properties in B(I). A complete norm topology stronger than the topology of poingwise convergence can only be one of uniform convergences. Arzela gave a necessary and sufficient condition of continuation of limit functions in sequences of continuous functions, among which topology can not be a norm topology.

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Available abstract

Pointwise convergence has no completeness in bounded function space B(I). The topology of uniform convergence has three characteristic properties in B(I). A complete norm topology stronger than the topology of poingwise convergence can only be one of uniform convergences. Arzela gave a necessary and sufficient condition of continuation of limit functions in sequences of continuous functions, among which topology can not be a norm topology.

Key concepts: Pointwise convergence, Mathematics, Topology (electrical circuits), Compact convergence, Product topology, Network topology, Norm (philosophy), General topology

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