| 1. | Alternatively, it can be argued using the uniform boundedness principle.
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| 2. | This may be taken as an alternative definition of total boundedness.
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| 3. | See the discussion on the " boundedness problem " below.
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| 4. | Thus, boundedness of on its domain does not imply boundedness of.
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| 5. | Thus, boundedness of on its domain does not imply boundedness of.
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| 6. | Hence these two theorems imply the boundedness theorem and the extreme value theorem.
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| 7. | For infinity, the result is a corollary of the uniform boundedness principle.
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| 8. | He discovered the Koecher boundedness principle in the theory of Siegel modular forms.
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| 9. | Note that compactness depends only on the topology, while boundedness depends on the metric.
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| 10. | The properties of total boundedness mentioned above rely in part on the axiom of choice.
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