Abstract
In studying the topology of infinite-dimensional spaces that are of particular interest in analysis and algebra, it has been customary to treat the spaces, where possible, as topological linear spaces and to use methods involving linear analysis and convexity. This chapter discusses certain topological linear spaces in compact spaces and considers special methods applicable to compact spaces, particularly, infinite product spaces. It describes the establishment of homeomorphisms among spaces and also the extensions of existing homeomorphisms defined on closed subsets. The principal specific infinite-dimensional spaces will be (l) Hubert space, l2, the space of square summable sequences with the norm topology, (2) the countable infinite product of lines, s, and (3) the Hubert cube or parallelotope, l∞, the countable infinite product of closed intervals. The complete topological classification of infinite-dimensional separable Fréchet spaces means that many further homeomorphism questions concerning such spaces are reduced to homeomorphism questions concerning any model of such spaces.
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