Averaging distances in certain Banach spaces
1997 ◽
Vol 55
(1)
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pp. 147-160
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Keyword(s):
Let E be a Banach space. The averaging interval AI(E) is defined as the set of positive real numbers α, with the following property: For each n ∈ ℕ and for all (not necessarily distinct) x1, x2, … xn ∈ E with ∥x1∥ = ∥x2∥ = … = ∥xn∥ = 1, there is an x ∈ E, ∥x∥ = 1, such thatIt follows immediately, that AI(E) is a (perhaps empty) interval included in the closed interval [1,2]. For example in this paper it is shown that AI(E) = {α} for some 1 < α < 2, if E has finite dimension. Furthermore a complete discussion of AI(C(X)) is given, where C(X) denotes the Banach space of real valued continuous functions on a compact Hausdorff space X. Also a Banach space E is found, such that AI(E) = [1,2].
Keyword(s):
1971 ◽
Vol 23
(3)
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pp. 468-480
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1985 ◽
Vol 97
(1)
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pp. 137-146
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Keyword(s):
1997 ◽
Vol 39
(2)
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pp. 227-230
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1962 ◽
Vol 14
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pp. 597-601
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1975 ◽
Vol 19
(3)
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pp. 291-300
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1989 ◽
Vol 32
(3)
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pp. 483-494
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Keyword(s):
1965 ◽
Vol 5
(4)
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pp. 453-462
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Keyword(s):
2000 ◽
Vol 130
(6)
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pp. 1227-1236
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