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- W1976197129 abstract "A bounded linear transformation T of a vector space S3 into itself will be called bicontinuous if the inverse of T (defined, of course, over the entire space) exists and is continuous. The group of bicontinuous transformations is dominant in any study of equivalence of transformations; that is, in the determination of the conditions implying relations of the type B = T~A T. The results of this paper constitute an analysis of the structure of bicontinuous transformations in certain spaces. Previous results of the author f indicate that the analysis here given is of a nature which will be useful in the equivalence problem, not only for transformations A which have a simple spectrum but also in more general situations. First the notion of a basis in a normal linear vector space 33 % is considered. A sequence of vectors { n), n £ S3, is customarily said to form a basis for S3 (necessarily separable) if for every ƒ e S3 there exists a unique sequence of numbers {an} such that/=2i°° n n.§ This definition is open to two objections: I t involves the notion of order of the elements of the basis; and for that reason it does not permit of immediate extension to nonseparable spaces. A new definition of basis is given below which is free of these objections. Proceeding in what is apparently another direction, the notion of complete orthonormal set in Hubert space (these sets form bases) may be extended to spaces S3. Thus after modifying suitably the idea of orthogonality and normality, one obtains a set. It is proved below that the concepts of heterogonal set and of basis are equivalent. In the second part, we deal exclusively with complex euclidean spaces S;|| briefly, these spaces are either unitary spaces, Hubert spaces, or nonseparable spaces of the Hilbert space type. The principal result is that if {#«} and {xf/a} are bases (or heterogonal sets) in @, then there exists a bicontinuous transformation T such that T <x=l/ot. Since the T transform of a basis is a basis (see below), the" @default.
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- W1976197129 date "1939-01-01" @default.
- W1976197129 modified "2023-10-13" @default.
- W1976197129 title "Bicontinuous linear transformations in certain vector spaces" @default.
- W1976197129 doi "https://doi.org/10.1090/s0002-9904-1939-07035-3" @default.
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