| 41. | Conversely, every bounded sequence in a Hilbert space admits weakly convergent subsequences ( Alaoglu's theorem ).
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| 42. | Either has a weakly Cauchy subsequence, or it admits a subsequence equivalent to the standard unit vector basis of.
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| 43. | Either has a weakly Cauchy subsequence, or it admits a subsequence equivalent to the standard unit vector basis of.
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| 44. | Any random sequence of digits contains arbitrarily long subsequences that appear non-random, by the infinite monkey theorem.
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| 45. | With each iteration, the neighborhood decreases, which forces a subsequence of iterates to converge to a cluster point.
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| 46. | They further show how to report " all " the longest increasing subsequences from the same resulting data structures.
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| 47. | The sequence of left neighbors may be found by an algorithm that maintains a stack containing a subsequence of the input.
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| 48. | A set in a Banach space is relatively weakly compact if and only if every sequence in has a weakly convergent subsequence.
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| 49. | An equivalent definition is that every sequence of points must have an infinite subsequence that converges to some point of the space.
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| 50. | Indeed, the LCS problem is often defined to be finding " all " common subsequences of a maximum length.
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