Category Archives: Logic

The unit ball of Hilbert space is not compact (p18 (B))

Show that the unit ball of Hilbert space is not compact. Proof. Let .  For each , define by , if , and .  Let . Now, let .  Then , where , and . Now, for each , for … Continue reading

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Product of a sequence of completely metrizable spaces is completely metrizable (p13 (C))

The product of a sequence of completely metrizable spaces is completely metrizable. Proof. Let be a sequence of completely metrizable spaces.  For each , let denote a compatible metric. Let have the product topology, which is given by the metric … Continue reading

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Continuous distance (p13 (E))

Let be a metric space.  Let .  Let by . Then is continous. Proof. Let .  Let be a basic neighborhood of in .  Then , for some . Let , .  Then . Also, , so . Therefore, is … Continue reading

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